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Non-revisiting paths and cycles in polyhedral maps

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NON-REVISITING PATHS AND CYCLES IN POLYHEDRAL MAPS


By

HARI PULAPAKA











A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY


UNIVERSITY OF FLORIDA


1995



















199











ACKNOWLEDGEMENTS


First and foremost, I would like to thank Professor Vince for his invaluable help

and advice during the preparation of this dissertation. He has spent countless hours

helping me understand the subtleties of my work. Sifting through some of the first

drafts must have been quite painful, and for this, I remain deeply appreciative of him.

Also, I would like to thank Professors Alladi, Davis, Mair, and White for taking the
time to serve on my supervisory committee.

Finally, I would like to dedicate this work to my parents and Cynthia. With-

out their moral support, this endeavor would be impossible and more importantly,

meaningless.












TABLE OF CONTENTS



ACKNOWLEDGEMENTS ............................ ii

LIST OF FIGURES ................................ iv

N O TATION . . . . . . . . . . . . . . . . . . vi

ABSTRACT . . . . . . . . . . . . . . . . . . vi

CHAPTERS
1 INTRODUCTION ............................... 1

1.1 Basic Definitions .. ...... ..... .. .. .. ..... ..... 1
1.2 Some History and Motivation ...................... 3
1.3 Non-Revisiting Paths and Cycles .................... 5
1.4 A Summary of the Research ....................... 7

2 NON-REVISITING PATHS ON SURFACES ................. 10

2.1 Non-Revisiting Paths on the Projective Plane, Torus, and Klein Bottle 11
2.2 Counter-Examples to the Non-Revisiting Path Conjecture ........ 29

3 NON-REVISITING CYCLES ON SURFACES ..................... 35

3.1 Polygonal Representation of Polyhedral Maps ................ 35
3.2 Polygonal Representation and Non-Revisiting Cycles ............ 46
3.3 A Graph-Coloring Problem and Non-Revisiting Cycles ........... 52

4 CONCLUSION ........ ................................. 60

REFERENCES ........ ................................... 61

BIOGRAPHICAL SKETCH ................................... 63












LIST OF FIGURES


Figure

1 A simpler proof of a result due to Barnette ......................... 11
2 A non-planar revisit of F to F on the projective plane .............. 15
3 A non-planar revisit of F1 to F on the projective plane ............. 16
4 The two possibilities for a non-planar revisit of F to F where F meets
the boundary of the annulus in the case of the torus ................ 18
5 The six possibilities for a non-planar revisit of F1 to F in the case in
F igure 4a ....................................................... 20-21
6 The three possibilities for a non-planar revisit of F1 to F in the case in
F igure 4b .......................................................... 23
7 A non-planar revisit of F to F where F does not meet the boundary of
the annulus ........................................................ 24
8 A non-planar revisit of F1 to F in the case in Figure 7 ............. 25
9 The two possibilities for a non-planar revisit of F2 to F2 in the case in
F igure 8 ........................................................ 26-27
10 A non-planar revisit of F to F in the case where both vertices are in
the interior of the annulus ......................................... 28
11 The five possibilities for a non-planar revisit of F to F in the case in
Figure 4a for the Klein bottle ................................... 30-31
12 The faces that constitute a counter-example to the non-revisiting path
conjecture for polyhedral maps on S2 ............................... 33
13 An orientation on the faces in Figure 12 that shows that the surface is
orientable ......................................................... 34
14 The faces that constitute a counter-example to the non-revisiting path
conjecture for polyhedral maps on N4 .............................. 36
15 An improper matching of edges on OP ............................ 40
16 An example of a polyhedral map M, a polygonal representation P of
M and the type of M ............................................. 41
17 Two polygonal representations of a polyhedral map on the torus... 42
18 The types of polygonal representations of polyhedral maps on the
projective plane, torus, and Klein bottle ......................... 43-44

iv











19 The two possibilities for a 2-connected, planar map with 14 vertices
and 3 hexagonal faces ............................................. 48
20 Polygonal representations for the polyhedral maps in Figures 12 and
14 ................................................................. 4 9
21 Non-planar, non-revisiting cycles on the projective plane .......... 52
22 Non-planar, non-revisiting cycles on the torus ..................... 53
23 Non-planar, non-revisiting cycles on the torus ..................... 54
24 Non-planar, non-revisiting cycles on the Klein bottle .............. 54
25 Non-planar, non-revisiting cycles on the Klein bottle .............. 55
26 Non-planar, non-revisiting cycles on the Klein bottle .............. 56
27 The graph-coloring conjecture is true for K3,3 ..................... 59
28 The boundary graphs of polyhedral maps on the projective plane, torus,
and K lein bottle ................................................... 62











NOTATION


G A finite graph.
V(G) The vertex set of G.
E(G) The edge set of G.
S A surface.
X The Euler Characteristic of a surface.
OS The boundary of the surface S.
S_ A surface homeomorphic to the connected sum of g tori.
Nk A surface homeomorphic to the connected sum of k projective planes.
M A polyhedral map.
M* The dual poyhedral map.
P A polygonal representation of a polyhedral map.
Tp The type of a polygonal representation P.











Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy




NON-REVISITING PATHS AND CYCLES IN POLYHEDRAL MAPS

By

Hari Pulapaka

August 1995

Chairman: Dr. Andrew Vince
Major Department: Mathematics

In this dissertation, three problems are considered. The first problem is related to
one of the most famous unsolved problems in the combinatorial theory of polytopes

called the Hirsch conjecture, which proposes a bound on the diameter of the graph of

a polytope. An equivalent conjecture due to Klee and Wolfe, called the non-revisiting

path conjecture, asserts that any two vertices of a polytope can be joined by a path

that does not revisit a facet. The non-revisiting path conjecture can be extended to

cell complexes that are more general than those that are the boundary complexes

of polytopes. In this regard, the non-revisiting path conjecture is known to be true

for polyhedral maps on the 2-sphere, projective plane, torus, and Klein bottle. In

this research, an elementary, unified proof of the validity of the non-revisiting path

conjecture for polyhedral maps on the projective plane, torus, and Klein bottle is
given. In addition, it is shown that for polyhedral maps, the non-revisiting path

conjecture is false for all other surfaces except possibly surfaces homeomorphic to the

connected sum of three projective planes.











The second problem concerns the existence of non-planar, non-revisiting cycles in
a polyhedral map on a surface. By results due to Barnette, it is known that every

polyhedral map on the projective plane, torus, and Klein bottle contains a non-

planar, non-revisiting cycle. In this regard, the notion of a polygonal representation
of a polyhedral map is introduced. This is analogous to the notion of representing

a surface as a polygon in the plane with the directed sides of the polygon matched

in pairs, except in this case, the representation preserves the combinatorial structure

of the underlying graph of the polyhedral map. Some properties of polygonal rep-

resentations are proved. As an application, an elementary, unified proof of results

due to Barnette involving the existence of non-planar, non-revisiting cycles on the

projective plane, torus, and Klein bottle is given.
The third problem is a graph-coloring conjecture that is shown to be true for

all planar graphs and K3,3. As an application, it is shown that any polyhedral

map on a surface homeomorphic to the connected sum of three projective planes, or
the connected sum of two tori, that has a non-separating polygonal representation,

contains a non-planar, non-revisiting cycle. This extends Barnette's result stated in

the second problem.












CHAPTER 1
INTRODUCTION



1.1 Basic Definitions




A polyhedron is the intersection of a finite collection of closed half-spaces in n-

dimensional Euclidean space, and a polytope is a bounded polyhedron. Equivalently,

a polytope is the convex hull of a finite set of points in Euclidean space. If a polytope

is d-dimensional, then we say that it is a d-polytope. A face of a polytope P is

0, P itself, or the intersection of P with a supporting hyperplane. With prefixes
denoting dimension, the 0-, 1-, and (d 1)-dimensional faces of a d-polytope are

called vertices, edges, and facets. A d-polytope P is simple if each of its vertices are

incident to precisely d edges, or equivalently, to d facets. A d-simplex is the convex

hull of d + 1 affinely independent points, and a polytope is called simplicial if each of

its facets is a simplex. There is a duality between the notions of simple and simplicial

polytopes. That is to say, there is a bijection between the set of simple d-polytopes

with n vertices and the set of simplicial d-polytopes with n facets that preserves

incidences and complements dimensions.

A graph G is a finite non-empty set of objects called vertices together with a

(possibly empty) set of unordered pairs of distinct vertices called edges. The vertex
set of G is denoted by V(G), while the edge set of G is denoted by E(G). A graph is

d-connected if the removal of fewer than d vertices yields neither a disconnected graph

nor the trivial graph. A cut-vertex of a graph is a vertex whose removal disconnects

the graph. Thus a graph is 2-connected if and only if it has no cut-vertices. The











graph of a polytope P is the one-dimensional skeleton of P. In particular, a theorem

of Steinitz and Rademacher [20] states that a graph is (isomorphic to) the graph of

a 3-polytope if and only if it is planar and 3-connected. By a generalization due to

Balinski [2], the graph of a d-polytope is d-connected. A directed graph consists of

a set of vertices and a set of ordered pairs of distinct vertices called directed edges.

A directed edge (u, v) is represented by an edge with endpoints u and v and an

arrowhead pointing towards v denoting the "direction" of the directed edge (u, v).

A surface S is a connected 2-dimensional manifold, possibly with boundary 9S.

There are two kinds of closed surfaces, orientable and non-orientable. The 2-sphere,

torus, double torus, and so on are orientable while the projective plane, Klein bottle

and so on are non-orientable. It is well known that any orientable surface may be

obtained by attaching a suitable number of handles to the sphere, while any non-

orientable surface may be obtained by attaching a suitable number of M6bius bands

to the sphere. An orientable surface denoted by Sg is said have genus g, if one must

add g handles to the sphere to obtain its homeomorphism type. On the other hand,

a non-orientable surface denoted by Nk is said to have crosscap number k, if one must

attach k M6bius bands to the sphere to obtain its homeomorphism type. If S and

S2 are surfaces without boundary, then their connected sum is the surface obtained

by removing the interior of a disk from S1 and S2 and then identifying the resulting

boundary components. Thus, the surface S. is homeomorphic to the connected sum

of g tori, while the surface Nk is homeomorphic to the connected sum of k projective

planes. Let G be a connected graph embedded on a surface S such that Gn aS = aS.

Then the pair (G, S) is called a map on S and is denoted by M. The vertices and edges

of M are those of G, and the faces of M are the closures of the connected regions in

the complement of G on S. If G is embedded on the plane, then the map is called a

planar map. A planar map has exactly one unbounded face. If M = (G, S) is a map











on a surface without boundary, then the dual map of M, denoted by M* is defined

as follows: for each face f of M, place a vertex f* in its interior. Then, for each edge

e in G, draw an edge e* between the vertices just placed in the interiors of the faces

containing the edge e. The resulting graph with vertices f* and edges e* is called the

dual graph of G, denoted by G*, and the resulting map (G*, S) is the dual map of
M. On the other hand, if S has boundary aS, then define the dual map as follows:

for each face f of M, place a vertex f* in its interior. Then, for each edge e in G not

belonging to 0S, draw an edge e* between the vertices just placed in the interiors of

the faces containing the edge e. The resulting graph G* is called the dual graph of G

and the resulting map is called the dual map of M. An important property satisfied

by 3-connected graphs embedded on the 2-sphere is that any two faces intersect on a

single edge, a single vertex or not at all. Faces that meet in this way are said to meet

properly. If all the faces are simply-connected and all faces meet properly, then the

map M is called a polyhedral map on the surface. A consequence of all faces meeting

properly is that every vertex of a polyhedral map has degree at least three. By a

result of Barnette [4], every polyhedral map is 3-connected, generalizing Steinitz's

Theorem.



1.2 Some History and Motivation




The feasible region of any non-empty linear programming problem is a polyhedron,
and conversely, given a polyhedron P, it is always possible to construct a linear

program with P as its feasible region. Edge-following algorithms, like the Simplex

algorithm, start with a vertex of the feasible region and traverse along successive

edges of the region according to some prescribed rule, until an optimum vertex is











reached. The d-step conjecture and its relatives (including the Hirsch conjecture)

play a crucial role in the study of the computational complexity of such edge-following

algorithms. The d-step conjecture, formulated by W. M. Hirsch in 1957 and reported

in 1963 by Dantzig in his book Linear Programming And Extensions [9], has several

equivalent forms. One version, dealing with the maximum diameter A(d, n) of (the

graphs of) d-dimensional polytopes with n facets, asserts that A(d, 2d) = d for all

d while the Hirsch conjecture asserts that A(d, n) < n d for all n > d > 2. It

was proved by Klee and Walkup [16] that the d-step and the Hirsch conjectures are

equivalent, though not necessarily on a dimension to dimension basis. The distance,

5e(u, v) between two vertices u and v of a polytope P is a lower bound on the

complexity of applying an edge-following algorithm to P with initial vertex u and

target vertex v. Thus A(d, n) is a lower bound for the worst-case behaviour of edge-

following LP algorithms over all d-polytopes with n facets. Since this applies to

all edge-following algorithms, A(d, n) estimates the worst possible behaviour of the

best possible edge-following algorithm. The d-step and Hirsch conjectures remain

unsettled, though they have been proved in many special cases, and counterexamples

have been found for slightly stronger conjectures. Specifically, the d-step conjecture

has been proved for d < 5. Although sharper results are known for small values of d

and n d, the best known general bounds for A(d, n) are due to Adler[l] and Kalai

and Kleitman [12], respectively. They are as follows :

(n-d) lod+
Ln -d 1]J + I < A(d, n) < n


It is generally believed that the d-step and Hirsch conjectures are false. However,

finding counterexamples to that effect would be merely a small first step in the line of

investigation related to the two conjectures. For the recent status of the conjectures











and their relatives, the survey paper by Klee and Kleindschmidt [17] provides an

excellent source.



1.3 Non-Revisiting Paths and Cycles




If F is a path in a polyhedral map M, a revisit of F to a face F is a pair of vertices

(x,y) such that Fi[x, y] n F = {x,y} where F[x,y] is the path along r from x to y.

Let (x, y) be a revisit of a path F to a face F. If the two paths along F from x to y

are denoted as F[x, y] and F[x, y], then the revisit (x, y) is said to be planar if either

F[x, y] U F[x, y] or F[x, y] U F[x, y] bounds a cell on the surface. (Note that if one

does then so does the other.). A path is non-revisiting if it has no revisits.

In research on the d-step and Hirsch conjectures, it has been found that the

conjectures can be stated in several equivalent forms (even though no solution to

any one of them seems to be in sight!). One equivalent formulation is in terms of

the existence of non-revisiting paths in the graphs of convex polytopes. Part of this

research is related to the following non-revisiting path conjecture of Klee and Wolfe

(also called the W, conjecture): Any two vertices of a polytope P can be joined by

a path that does not revisit any facet of P. Despite an apparent greater strength of

this conjecture (which prompted its original formulation), it is known [15] that the

non-revisiting path conjecture is equivalent to the Hirsch conjecture.

If P is a 3-polytope, then the faces of P form a polyhedral map on the 2-sphere

and the validity of the non-revisiting path conjecture along with some strengthened

forms of the non-revisiting path conjecture are proved [3,14,15]. Klee [13] conjectured

that the non-revisiting path conjecture might be true for cell complexes that are more

general than the boundary complexes of convex polytopes. In this regard, Larman











[18] has shown that the conjecture is false for a very general type of 2-dimensional

complex. Mani and Walkup [19] have shown that the conjecture is false for 3-spheres.

Barnette [5,7] has recently shown that the non-revisiting path conjecture is indeed

true for polyhedral maps on cell complexes that are homeomorphic to the projective

plane and the torus. Engelhardt [10] has shown in her Ph.D. dissertation that the

non-revisiting conjecture is also true for polyhedral maps on the Klein bottle. In a

recent paper [8], Barnette gives counterexamples to the non-revisiting path conjecture

that are polyhedral maps on the surfaces S8 and N16.

Similar to the notion of a path in a polyhedral map having a disconnected inter-

section with a face of the polyhedral map, one may consider a cycle in the underlying

graph of a polyhedral map that has a disconnected intersection with a face of the

polyhedral map. A cycle of a polyhedral map refers to a cycle in the underlying graph

of the polyhedral map. Let M = (G, S) be a polyhedral map and C be a cycle in M.

Then C is said to be non-planar if it does not bound a cell on S. Suppose R[s, t] is

a path along C from s to t such that for some face F, s and t are on F, and R[s, t]

along with either path along F from s to t bounds a cell on S. Then the path R[s, t]

is called a planar revisit of C to the face F. A cycle is non-revisiting if it does not

have any revisits; in other words, for each face F of M, C n F is either empty, or

connected. According to a theorem due to Barnette [6], if M has a non-planar cycle
all of whose revisits are planar, then M has a non-planar, non-revisiting cycle. It is

also known that every polyhedral map on a projective plane, torus, or Klein bottle

has a non-planar, non-revisiting cycle [6]. However, the problem of the existence of

such cycles on other surfaces is still open.

Using the result for the three surfaces mentioned above, Barnette [6] proves that

every polyhedral map on the torus is the union of two face-disjoint subcomplexes











that are annuli. Similar decomposition theorems are proved for the projective plane

and Klein bottle.



1.4 A Summary of the Research




Primarily, three problems are considered in this dissertation. The first problem

is related to the non-revisiting path conjecture due to Klee and Wolfe. In its original

formulation, the non-revisiting path conjecture was in the context of convex polytopes.

A generalization due to Klee [13] of the conjecture led to the problem of the existence

of non-revisiting paths between any two vertices of a polyhedral map on a surface.

Chapter 2 deals with this question. Specifically, in Section 2.1, an elementary, unified

proof of the non-revisiting path conjecture for polyhedral maps on the projective

plane, torus, and Klein bottle is given. Although these results are already known,
the earlier proofs for the three surfaces are quite different. The proof given here uses

a result due to Barnette, which states that any polyhedral map on the projective

plane, torus and Klein bottle has a non-planar, non-revisiting cycle. Furthermore, in

the case of the torus and Klein bottle, cutting along this non-revisiting cycle yields an

annulus. Hence in the case of the projective plane, the polyhedral map is cut along a

non-revisiting cycle yielding a cell whose boundary corresponds to the non-revisiting

cycle and the arguments presented pertain to the cell thus obtained. In the case of the

torus and Klein bottle, the arguments pertain to an annulus whose bounding cycles

correspond to the the non-revisiting cycle in the polyhedral map. The unification

of the proofs for the three surfaces is obtained by considering the same basic cases

for all three surfaces, namely either both vertices lie on the non-revisiting cycle, one
lies on the non-revisiting cycle and one does not, or neither of the two vertices lie on











the non-revisiting cycle. The proof also utilizes an important lemma due to Barnette

which states that a path with only planar revisits can be modified to a non-revisiting

path. This result plays a key role even in the earlier proofs. In Section 2.1, a simpler

proof of this lemma is given. Section 2.2 deals with the non-revisiting path conjecture

for the other surfaces. In Engelhardt's dissertation, a proof of the validity of the non-

revisiting path conjecture for the surface S2 is given. In Section 2.2, it is shown that

this is impossible! In fact, it is shown that the non-revisiting path conjecture is false

for polyhedral maps on the surfaces Sg, g > 2, and Nk, k > 4. Thus, the non-revisiting

path conjecture for polyhedral maps is settled for all surfaces except N3.

The second problem concerns the existence of a non-planar, non-revisiting cycle

in a polyhedral map. As stated earlier, the only surfaces that are known to contain

such cycles are the projective plane, torus, and Klein bottle [6]. In this context, in

Section 3.1, the notion of a polygonal representation of a polyhedral map is defined.

This is analogous to that of representing a surface as a polygon whose sides are

directed and matched in pairs. Except, in the case of a polygonal representation of

a polyhedral map, the representation preserves the combinatorial structure of the

underlying graph of the polyhedral map. In other words, a polygonal representation

is a polyhedral map on a closed disc with certain matching conditions on the edges

of the polyhedral map that lie on the boundary of the disc. It is shown that every

polyhedral map has a polygonal representation. Next, the notion of a non-separating

polygonal representation of a polyhedral map is defined. As will be evident from its

definition, the existence of such a representation is a rather desirable property of
a polyhedral map. An interesting question is: Which polyhedral maps have a non-

separating polygonal representation ? In this regard, it is shown that there exist an

infinite family of polyhedral maps that do not possess a non-separating polygonal

representation. Elementary Euler Characteristic arguments allow the enumeration of











all the types of polygonal representations of polyhedral maps on the projective plane,

torus, and Klein bottle. In Section 3.2, an elementary, unified proof of Barnette's

result [6] on non-revisiting cycles on the projective plane, torus, and Klein bottle for

polyhedral maps that possess a non-separating polygonal representation is given.
Motivated by the second problem, a graph-coloring conjecture is proposed in

Section 3.3. This is the third problem considered. It is shown that if the graph-
coloring conjecture is true in a special case, then every polyhedral map that has a

non-separating polygonal representation, in fact, has a non-planar, non-revisiting cy-
cle. It is shown that the graph-coloring conjecture is true for all graphs that contain

a triangle, all planar graphs, and K3,3. As a consequence, it follows that every poly-

hedral map on a surface homeomorphic to Nk, k = 1, 2, 3, and S9, g = 1, 2 that has a

non-separating polygonal representation, contains a non-planar, non-revisiting cycle.

This extends Barnette's result [6] to this class of polyhedral maps on the surfaces N3

and S2.












CHAPTER 2
NON-REVISITING PATHS ON SURFACES




This chapter deals with the non-revisiting path conjecture for polyhedral maps. In

Section 2.1, the non-revisiting path conjecture is shown to be true for polyhedral maps

on the projective plane, torus, and Klein bottle. Although these results are already

known, the earlier proofs due to Barnette [5,7] and Engelhardt [10], are different. In

Section 2.1, a simpler, unified proof for all three surfaces is provided. In the case of
the projective plane, by a result due to Barnette [6], the surface is cut along a non-

planar, non-revisiting cycle to yield a cell and the proof consists of the considering

the following three cases:

1. Both vertices involved lie on the boundary of the cell.

2. One vertex lies on the boundary of the cell while the other lies in the interior of

the cell.

3. Both vertices lie in the interior of the cell.
In the case of the torus and Klein bottle, the surface is cut along a non-revisiting

cycle in the polyhedral map yielding an annulus. The unification of the proof for all

three surfaces is achieved by considering the same three cases stated above.

In Section 2.2, the non-revisiting path conjecture is settled for polyhedral maps

on all the remaining surfaces except N3, the connected sum of three copies of the

projective plane (or equivalently, the connected sum of the torus and the projective

plane, or the Klein bottle and the projective plane). Specifically, it is shown that the

non-revisiting path conjecture is false for all the remaining surfaces, except possibly

N3 and counter-examples are provided to this effect.











2.1 Non-Revisiting Paths on the Projective Plane, Torus, and Klein Bottle




Although the proof of the validity of the non-revisiting path conjecture for poly-

hedral maps on the projective plane, torus, and Klein bottle given here has many

details, the ideas involved are quite elementary.
First, a simpler proof of an important lemma originally due to Barnette [5] is

presented.


Lemma 2.1.1 Let M be a polyhedral map with vertices u and v. If there is a path

in M joining u and v all of whose revisits are planar, then there is a non-revisiting

path between u and v.

Proof. Let F[u, v] be a path in M all of whose revisits are planar. If F[u, v] is not

a non-revisiting path, then there is a vertex x on F[u, v] with the following properties:

(1) There is a non-revisiting path Po[u, x] between u and x.

(2) The path Fo[u, x] U F[x, v] has only planar revisits.

A path satisfying (1) and (2) exists; simply take x = u.
(3) Among all choices for x satisfying (1) and (2), choose the one which is furthest

along the path F[u, v].



.... .. .......... .. .. . .. .


r x Y


Figure 1. A simpler proof of a result due to Barnette.











If x = v, we are done, otherwise we will obtain a contradiction. Let (z, Y) be a
revisit of the path Fo[u, x] U F[x, v] to a face F of M. By statement (3), z E Fo[u, x]

and Y E F[x, v]. Among all revisits by this path we choose F so that z is nearest to
u along Fo[u, x]. Now consider the path Fi = F0[u, z] U F[z, y] u r[y, vi from u to v

(indicated by the dotted path in Figure 1) and observe the following :

(i) F1 is a path from u to v all of whose revisits are planar. To see this note that
F0 itself has no revisits. A revisit involving vertices of F[y, v] alone has to be planar

since F has only planar revisits. A non-planar revisit by 1 cannot involve vertices
of F[z, y] since the closed path F[z, y] U F[y, x] U Fo[x, z] bounds a cell. Finally, if a

revisit by F1 involves a vertex of F and a vertex of F0, then it must be planar since

Fo[u, x] U F[x, v] admits only planar revisits.
(ii) F, [u, y] does not revisit any face of M. A revisit by F, [u, y] to a face F must

involve y and a vertex & of Fo[u, z). Note that : 7 z; otherwise F and F1 meet
improperly at y and z. Now (2, y) is a revisit of the path Fo[u, x] U F[x, v]. This

contradicts the choice of F with z nearest to u on Fo[u, x].

The existence of y contradicts the choice of x as the vertex that was furthest along

F[u, v] satisfying conditions (1) and (2). 0


Lemma 2.1.2. Let S be a surface with boundary aS and M = (G, S) a polyhedral

map on S such that the intersection of any face of M with aS is either empty, or

connected. Then any two vertices of M that lie in the interior of S can be joined by
path in M that is contained in the interior of S.
Proof : Since G is connected, there is a path F from u to v in M. If F lies in

the interior of S, we are done; so assume that F n aS 5 0 and let H = F[x, y] be a
connected component of F n aS with the order of vertices along F being u, x, y, v. Let
x' and y' be the vertices of H that are incident to x and y, respectively. It is possible











that some of x, x', y', and y are the same vertex. Denote by F,,, ..., Fm, m >_ n the

faces of M that meet H[x', y'] but not x or y. With x" as the vertex of F not on H that

is incident to x, let F1, ..., F,-, be the faces of M that meet H and lie in the region

determined by the sector with central angle x"xx'. Likewise, let Fm+i, ..., FN be the

faces that meet H and lie in the region determined by the sector with central angle

y'yy", where y" is the vertex of f not on H that is incident to y. For i = 1,..., N 1,

let xi be the vertex on the edge Fi f Fi+l that doesn't belong to H. Choose x0 on

F[u, x) with the property that for some s = 1, ..., N, the face F, contains an edge

not in F that is incident to x0. Such a vertex exists for otherwise one of the F's has

a disconnected intersection with OS, which is a contradiction. Similarly, choose yo

on F(y, v] such that the face Ft, t > s, contains an edge not in F that is incident to

Yo- Note that x0 and yo must lie in the interior of S for otherwise F, or Ft has a

disconnected intersection with as (meeting aS at both H and xO or yo, respectively),

which is a contradiction. Construct a path from x0 to yo that doesn't meet OS as

follows:

Since x0 and x, belong to F, and lie in the interior of S, there must be a path

F[xo, x,] from x0 to x, along the face F, that avoids OS, for otherwise F, n as is

disconnected. For k = s, ..., t 2, the vertices Xk and Xk+1 lie on the face Fk+l and

are in the interior of S. Hence by the argument above, for each k, there is a path

Fk+l[xk, Xk+l] along Fk+j that lies in the interior of S. And let Ft[xt-,,yo] be the

interior path from xt-l to yo along Ft. By construction,
t-2]
I= F,[xo, xa] U U Fk+l[xk, xk+1] U F[xt-,,yo] n as = 0.
k=s

In F, replace F[xo, yo] by I. Now there may be repeated vertices on F. In this case,
remove the vertices of F that appear between successive occurrences of each repeated
vertex, eventually yielding a path F1 from u to v where F1 n H = 0. If r1 lies in











the interior of S, we are done; otherwise perform the same modification as above to

a connected component of F1 n aS. When it is no longer possible to perform any

modifications, the result must be a path from u to v that is contained in the interior

of S.


The following two lemmas are due to Barnette [6].


Lemma 2.1.3. Every polyhedral map on the projective plane has a non-planar,

non-revisiting cycle.


Lemma 2.1.4. Every polyhedral map M on the torus or Klein bottle contains a

non-revisiting cycle C such that cutting M along C yields an annulus.


We are now in a position to state and prove the main result of this section.


Theorem 2.1.1. Any two vertices of a polyhedral map M on the projective plane,

torus or Klein bottle can be joined by a non-revisiting path.

Proof : For each surface, we will show that any two vertices u and v can be

joined by a path in M all of whose revisits are planar. Consequently, by Lemma

2.1.1, there is a non-revisiting path joining u and v.

First consider the case where M is a polyhedral map on the projective plane. By

Lemma 2.1.3, M has a non-revisiting cycle C such that cutting M along C yields

a cell H whose boundary corresponds to the cycle C. Without loss of generality,

consider the following cases:

1. u and v lie on C. In this case, either of the two paths along C from u to v must

be non-revisiting (since C is non-revisiting).











2. u lies on C and v does not lie on C. Since every vertex of M has degree at least

three, there must be a vertex ul of M in the interior of the cell H such that uu1 is an
edge of M. Since the cycle C is non-revisiting, the intersection of any face of M with

OH is either empty, or connected. Hence by Lemma 2.1.2, there is a path F0 joining

ul and v in M that is contained in the interior of H. Define F = F0 U uul. Thus

F is a path joining u and v that meets the boundary of H in only u. If F has only

planar revisits, we are done by Lemma 2.1.1. It is clear that a non-planar revisit of

F to a face Fmust involve a vertex s lying on F(u, v] and u. Among all non-planar
revisits of F, choose F so that s is nearest to v along F. Replace F by the path

F1 F[u, s] u F[s, v] indicated by the dotted path in Figure 2.





t


Figure 2. A non-planar revisit of F to F on the projective plane.











Again, if F1 has only planar revisits, we are done. On the other hand, if I1 has

a non-planar revisit to a face F1, then it must involve a vertex s1 of F,(s, v] and a

vertex of Fl[u,t] as shown in Figure 3.



t t2 U


Figure 3. A non-planar revisit of F1 to F on the projective plane.





Among all choices for F1, choose the one for which s, is nearest to v along F1 and

let t2 be as shown. Replace F1 by the path F2 = Fl u, t2] U Fl[t2, su] U F[sl,V]. It

can be easily checked that F2 can have only planar revisits.

3. Neither u nor v lies on C. In this case, both u and v lie in the interior of the cell

H. Hence by Lemma 2.1.2, there is a path F joining u and v that is contained in the











interior of H. Such a path can have only planar revisits and by Lemma 2.1.1, we are

done.

Next consider the case where M is a polyhedral map on the torus. By Lemma

2.1.4, M has a non-revisiting cycle C such that cutting M along C yields an annulus

A. Let C1 and C2 be the bounding cycles of A corresponding to C and without loss

of generality, consider the following three cases :



1. u and v lie on C. In this case, either of the two paths along C from u to v is

non-revisiting (since C is non-revisiting).



2. u lies on C and v does not lie on C. As in the case of the projective plane, there

is a path F joining u and v that meets oA at only u. Without loss of generality,

assume that F meets C, at only u and avoids C2. If all of F's revisits are planar, we

are done. So assume that F has a non-planar revisit (s, t) to a face F with the vertex

s closer to v along IF than the vertex t is to v. Among all non-planar revisits of F

choose F so that s is nearest to v along F. Note that F cannot meet both C1 and C2

in A for this would mean that the cycle C revisits F which is a contradiction to the

assumption that C is a non-revisiting cycle. Consider the following two cases :



i. F contains u. Up to symmetry, there are two possibilities for F (depending on

whether F meets C1 or C2) as shown in Figure 4.















U S U






v C2 v









a. b.

Figure 4. The two possibilities for a non-planar revisit of F to F where
F meets the boundary of the annulus in the case of the torus.

First consider the case in Figure 4a. above. In this case, t = u. Let to be as

shown. Replace F by the path I', = F[u, s] U F[s, v] indicated by the dotted path

in Figure 4a. If F1 has only planar revisits, then we are done; so assume that F1

has a non-planar revisit (sl, t1) to a face F1. Note that s, and tj cannot both lie on
F1 [u, s] since this would mean that F and F meet improperly. Hence, without loss
of generality, assume that s, lies on Fl(s, v] and t1 lies on F' [u, s]. Among all choices

for F1, choose the one for which s, is nearest to v along F1. It is easy to see that it

suffices to consider the six possibilities for F shown in Figure 5.
























a. b.













































V






Figure 5. The six possibilities for a non-planar revisit of F1
to F in the case in Figure 4a.











In the cases in Figures 5a, 5b, and 5c, F has a non-planar revisit to the face F
contradicting the choice of F with s nearest to v along P. Consider the case in Figure

5d and let tj be as shown. Replace F1 by the path F2 = F1[u, ti] U F [t, S1] U Fl[sI, v]

as shown in Figure 5d. Now F2 has only planar revisits, Next consider the case in

Figure 5e and replace F1 by the path F2 = F[u, si] U Fi[si,v], where F[u, si] is a

path from u to s, along F that meets 9A at only u. Again, it can be checked that

r2 can have only planar revisits. And in the case in Figure 5f, replace F1 by the
path F2 = F1[u, t1] U Fl[ti,si] U F, [s,v] where Fl[ti,si] is a path along F1 that lies

in the interior of A. Now the only possibility for a non-planar revisit of F2 to a face

F2 is for it to involve a vertex s2 of F2[tl,sl] and a vertex t2 of F2[u, to]. Among

all such choices for F2, choose the one for which s2 is nearest v along F2. Now

r13 = F2[u, t2] U F2[t2, s2] U P2[52, v] can have only planar revisits.


Next consider the case in Figure 4b. Replace I by the path F1 = F[u, s] U F[s, v]

indicated by the dotted path in Figure 4b. Again, if 1P has only planar revisits, we are

done. So assume that r, has a non-planar revisit (s,, t1) to a face F1. Topologically,

there are three possibilities for the face F. Without loss of generality, assume that

the three possibilities for F1 are as shown in Figure 6.


In the cases in Figures 6a. and 6b, F has a non-planar revisit to F which con-

tradicts the choice of F with s nearest to v along P. Hence F must be as shown in

Figure 6c. Among all choices for F1, choose the one for which s, is nearest to v along

F1. Replace F1 by the path F2 = F1[u, t] U Fl[ti, sl] U F,[sI, v] where Fl[ti,s] is a

path along F from s, to tl that avoids C, and C2 except possibly meeting C1 at u

in the case where tl = u. Such a path exists for otherwise, C revisits F which is a

contradiction. It can now be checked that F2 can have only planar revisits.









22


Ul u













UU













.C
V





a. b.
























Figure 6. The three possibilities or a non-planar revisit of r
to F in the case in Figure 4b.











(ii) F does not contain u. Recall that F can meet at most one of C, or C2. First

consider the case where F does not meet C1. Since F does not contain u, there must

be path along F from s to t that is contained in the interior of A. Without loss of
generality, assume that F is as shown Figure 7.


Figure 7. A non-planar revisit of F to a F where F does not meet
the boundary of the annulus.



In this case, replace F by the path F1 = F[u, t] U F[t, s] U F[s, v] where F[t, s] is a

path along F from s to t that is contained in the interior of A. If F1 has only planar
revisits, we are done; so assume that F, has a non-planar revisit to a face F1. If this
non-planar revisit involves a vertex of Fr(s, v] and a vertex of Fi[u,t], then the proof
is identical to the one given for Figure 6c. It can be checked that the only possibility











is for the non-planar revisit to involve a vertex sl of F1 (s, t) and u as shown in Figure

8.


Figure 8. A non-planar revisit of F1 to F in the case in Figure 7.


Among all choices for F1, choose the one for which s, is nearest to v along 11

and replace 1 by the path F2 = F[u, sI] U r, [sl, v] as shown in Figure 8. Now there
are two possibilities for a non-planar revisit of F2 to a face F2. These are shown in

Figures 9a and 9b. In both cases, among all such choices for F2, choose the one for











which s2 (as shown in Figures 9a and 9b) is nearest to v along F2 and let t2 be as
shown. Next replace F2 by the path 173 = F2[u,t2] U F2[t2, s2] U F2[92, v]. It can be

checked that 173 can have only planar revisits.
The case where F does not meet C2 only is similar to the case above. If F neither

meets C1 nor C2, the proof is again similar to the one given above.

3. Neither u nor v lies on C In this case, u and v lie in the interior of the annulus A.

By Lemma 2.1.2, u and v can be joined by a path F that is contained in the interior

of A. The proof that r can be modified to a path joining u and v that has only

planar revisits is identical to the one for Figure 7. Thus in all cases, u and v can be

joined by a path in M that has only planar revisits and by Lemma 2.1.1, we are done.
U
t2 to




































Figure 9. The two possibilities for a non-planar revisit of r2
to FA in the case in Figure 8.


Figure 10. A non-planar revisit of F to F in the case where
both vertices are in the interior of the annulus.











Next consider the case where M is a polyhedral map on the Klein bottle and let

u and v be vertices of M. The proof in this case is similar to that in the case of

the torus with a few subtle differences. As before, use Lemma 2.1.4 to cut M along

a non-planar, non-revisiting cycle C yielding an annulus A with bounding cycles C1

and C2 and consider the following three cases:

1. u and v lie on C. In this case, the argument is identical to the one given above for

the torus.
2. u lies on C and v does not lie on C Without loss of generality, assume that u lies

on C1. As in the case of the torus, consider the path F joining u and v shown in

Figure 4. If F has only planar revisits, we are done, so assume that F has a non-planar
revisit (s, t) to a face F and among all non-planar revisits of F, choose the one for

which s is nearest to v along F. Consider the following cases :

i. F contains u It suffices to consider the cases shown in Figures 4a. and 4b. Replace
F by the path F, = F[u, s] U P[s, v]. If P has only planar revisits, we are done; so

assume that F, has a non-planar revisit to a face F. As in the case of the torus, such

a revisit must involve a vertex s, of F1(s, v] and a vertex of 1[u, s]. Without loss of

generality, there are five possibilities for F as shown in Figure 11. In the cases in

Figures hla, llb, and 11c, F has a non-planar revisit to F contradicting the choice

of F with s nearest to v. So consider the case in Figure 1ld and let t2 be as shown.

Replace I' by the path r2 = 1i[u,t2] U Fi[t2, si] U IF[sl,v]. It can be checked that

F2 can have only planar revisits. The case in Figure lie is similar to the analogous

case for the torus.
ii. F does not contain u. This case is similar to the analogous case for the torus.

3. Neither u nor v lies on C. Again, this case is similar to the analogous case for

the torus.

This concludes the proof of the theorem.








28


U u




















a. b.
U U





















C. d.











U







t0



V







e.

Figure 11. The five possibilities for a non-planar revisit of F to F
in the case in Figure 4a for the Klein bottle.



2.2 Counter-Examples to the Non-Revisiting Path Conjecture




Barnette has recently shown that there exist polyhedral maps on the surfaces

S8 and N16 for which the non-revisiting path conjecture is false. In Engelhardt's
dissertation, it is claimed that the non-revisiting path conjecture is also true for

polyhedral maps on the surface S2. In this section, we settle the non-revisiting path

conjecture for polyhedral maps on all surfaces except N3. Specifically, it is shown

that for each g _> 2, the non-revisiting path conjecture is false for the surface S, and

for each k > 4, the non-revisiting path conjecture is false for the surface Nk. This











of course contradicts Engelhardt's result for the surface S2. Since the non-revisiting

path conjecture is already known to be true for the 2-sphere, projective plane, torus,

and Klein bottle, the only surface for which the non-revisiting path conjecture is still

open is N3.


The Counterexamples. The polyhedral maps that constitute the counterexam-

ples for the surfaces mentioned above will be described in terms of the polygons that

form the faces of the polyhedral map. Thus, the vertices and edges of the polyhedral

map are those of the polygons and the surface is obtained by glueing the polygons

together along the edges with the same labels.

First consider the orientable case and let F1, ..., F16 be the polygons with the

vertex-labelling shown in Figure 12.



4 A 1 B 3 A

F F F5 F6
1 256
D B 4 2
x y
F F F F
3 4 7

3 C 2 C 1 D

C 1 D 3 D 2 A 4





3 A 1 B 4 B 2 C

Figure 12. The faces that constitute a counter-example to the
non-revisiting path conjecture for polyhedral maps on S2.











Paste the polygons together by identiying the edges with the same labels. It can

be checked that the result is a surface S without boundary with the map Mi given

by the union of the 16 polygons. Next, it is shown that S is orientable. This is done

as follows:

First note that each face has two possible directions for its boundary walk. Assign

an "orientation" to each face by choosing one of these two directions. If every face

can be assigned an orientation in such a way that adjacent regions induce opposite

directions on every common edge, the surface S is orientable. Such an orientation

for the faces Fi, ..., F16 is shown in Figure 13.


3 A


1 B


Figure 13. An orientation on the faces in Figure 12 that shows that
the surface is orientable.

Observe that Mi has 10 vertices, 28 edges, and 16 faces. Hence the Euler Char-

acteristic of S is -2. And since S is orientable, it must be homeomorphic to S2. It is


4 A 1



X



3 C 2


B 3


00

CYC
y











easy to check that the faces F1, ..., F16 meet properly. Hence M is a polyhedral map
on S2. It remains to be shown that M1 does not have the non-revisiting property.

We will show that the vertices labelled x and y cannot be joined by a non-revisiting

path in M1. The proof is by contradiction; so assume that F is a non-revisiting path

joining x and y in M1. Without loss of generality, assume that the vertex incident to

x along F is the vertex labelled A (the proof is symmetric in the other cases). Note

that in this case, the path F has left the faces labelled F3 and F4. Furthermore, the

label A also appears on the face F6. Since F was assumed to be non-revisiting and

the vertex y lies on F6, the remainder of F must lie on the face F6. There are two

ways of getting from A to y along F6, namely, through the vertices labelled 2 or 3. If

F passes through the vertex 2, then the face F4 is revisited by F, which contradict-

ing the assumption that F is a non-revisiting path. On the other hand, if F passes

through the vertex labelled 3, then F revisits the face F3; also a contradiction. Thus,

there can be no non-revisiting path from x to y in M.

In order to prove the result for the surface Sg, g >_ 3, we form the connected sum

of the surfaces S2 and S_2 as follows:

Let M, be a polyhedral map on the surface S_2 such that M has a triangular face

T. Assign the same labelling on the vertices of T as the face F9 of M1. Glue the

polyhedral maps M1 and M1' by identifying the faces F9 and T. Then remove this

face from the cell complex. The result is a map MI" on the surface S2. In fact, M"

is a polyhedral map on S.. In order to prove this, it suffices to show that the faces of

M," meet properly. Let F and G be faces of M". If F and G are also faces of M1,

or M,', then they clearly meet properly. Without loss of generality, assume that F is
a face of M1 and G is a face of M1'. The only way that they can meet in Mi" is if F

meets F9 in M1 and G meets T in M1'. In this case, F and G have to meet properly








33


for otherwise, either F and F9 meet improperly which is a contradiction since M is

a polyhedral map, or G and T meet improperly which is a contradiction since M1'

was chosen to be a polyhedral map. The proof that x and y cannot be joined by

a non-revisiting path in M1" is identical to the proof given earlier. Thus, MI" is a

polyhedral map on Sg without the non-revisting property.


Next, we show that the non-revisiting path conjecture is false for the surface N4.

In this case, consider the 17 polygons F1, ..., F17 with the vertex-labelling shown in

Figure 14.


4 A 1

F F
2

X
F F
3 4

3 C 2


D 2


Figure 14. The faces that constitute a counter-example to the
non-revisiting path conjecture for polyhedral maps on N4.

As in the case of the surface S2, paste the polygons together by identifying edges

with the same labels. Again, the result is a surface S without boundary with the

map M2 given by the union of the faces F1, ..., F17. It can be checked that the faces











of S cannot be assigned an orientation as described earlier in the proof. To check if

such an assignment is possible, first assign an arbitrary orientation to a particular

face. This forces an orientation of each face that shares a common edge with the

original face. Since the surface is connected, the process can be continued untill the

orientation of each face has been forced. Either the result is an orientation for the

embedding, or else the given embedding has no orientation in which case the surface

is non-orientable. Hence S is non-orientable. M2 has 11 vertices, 30 edges, and 17

faces. Thus S has Euler Characteristic -2 and must be homeomorphic to N4. Once

again, it can be checked that the faces F1, ..., F17 meet properly. Hence M2 is a

polyhedral map on N4. The proof that M does not have the non-revisiting property

is identical to the one given for the surface S2.

A counterexample for the surface Nk, k > 5 is obtained by glueing a polyhedral

map on Nk-4 to M2 as described in the orientable case. Again, this method yields a

counterexample for each surface Nk, k > 5.












CHAPTER 3
NON-REVISITING CYCLES ON SURFACES




This chapter consists of three sections. In Section 3.1, the notion of a polygonal
representation of a polyhedral map is introduced. As will be seen, this is a convenient

way to represent a polyhedral map as a polygon in the plane. It is shown that every

polyhedral map on a surface has such a representation and some useful properties

of polygonal representations are proved. The notion of a non-separating polygonal

representation is defined. An interesting question is: Which polyhedral maps have a

non-separating polygonal representation ? It is shown that not all polyhedral maps

have a non-separating polygonal representation. In Section 3.2, polygonal represen-

tations are used to provide a simple, unified proof of the existence of a non-planar,

non-revisiting cycle in a polyhedral map on the projective plane, torus, and Klein

bottle. This is done for polyhedral maps that have a non-separating polygonal rep-

resentation. And in Section 3.3, a graph-colouring problem that is motivated by the

question of non-planar, non-revisiting cycles in a polyhedral map, is considered. The

conjecture is shown to be true for all planar graphs and K3,3. Consequently, Bar-

nette's result [6] on the existence of the above mentioned cycles on the projective

plane, torus, and Klein bottle is extended to the surfaces N3, and S2.



3.1 Polygonal Representation of Polyhedral Maps




It is well known that any compact, connected surface may be represented as a

polygon in the plane with labeled and directed sides. The directed sides are matched











in pairs and the surface may be obtained by identifying the matched directed sides
of the polygon. Analogously, if M = (G, S) is a polyhedral map, then a polygonal
representation of M is a representation of M as a polygon in the plane that preserves

the combinatorial structure of G. Thus the sides of the polygon are in fact, edges in

G. This notion is made more precise below.
A polygonal map P is defined as a polyhedral map on a closed disc such that:

(1) The vertices of OP are labeled, and every label appears at least twice on OP.
(2) The edges of OP are directed and there is a matching on this set of directed edges

of OP that matches each directed edge labeled (A, B) with another directed edge

labeled (A, B) with the same labels.

If M is a polyhedral map on a surface, then a polygonal map P is called a polygonal

representation of M if
(1) M is obtained from P by identifying matched edges on OP and,
(2) after the identifications, each vertex label appears exactly once in M.

Note that, in general, a polyhedral map may have several polygonal representa-

tions. Figure 17 shows two polygonal representations of a polyhedral map (whose

underlying graph is K7) on the torus. Also if P is a polygonal representation of M,

then there can be no matched edges on OP as in Figure 15 below. Otherwise label

A either appears only once on OP, contradicting statement (1) in the definition of a

polygonal map, or label A appears more than once in M, contradicting statement (2)

in the definition of a polygonal representation. A polyhedral map and a polygonal

representation of the polyhedral map are shown in Figure 16.











A A


B B B B

Figure 15. An improper matching of edges on OP.

Theorem 3.1.1. Every polyhedral map M, not on the sphere, has a polygonal

representation.

Proof. Label the vertices of M. Since the underlying graph of the dual map M*

is connected, it has a spanning tree T*. There is a bijection between the edges e in

E(M) and the edges e* in E(M*). Here e* is the unique edge that crosses e. Let E*

denote the complement of T* in M* and define E = {e G E(M)lc* E E*}. Cut M

along the edges in E. Since T* is planar, the result is a planar map P that satisfies

all the conditions for it to be a polygonal map except condition (1). If a pair of edges

are matched as in Figure 15, then glue them back together. Now every vertex label

on OP appears at least twice on 9P and the map still remains planar. Furthermore,

P is a map that satisfies all the conditions for it to be a polygonal representation of

M. 0

Let P be a polygonal map and assume that a pair of directed edges (A, B) and

(B, C) on OP are incident at B. Further assume that the respective matching edges

(A, B)' and (B, C)' are also incident at B. Replace (A, B) and (B, C) by a single

directed edge (A, C); similarly replace (A, B)' and (B, C)' by a single directed edge

(A, C)'. Call such a replacement a concatenation. Perform concatenations along aP

until it is no longer possible to do so. Call OP with the resulting vertex labeling

the type of P, denoted by Tp. Figure 16 shows a polyhedral map M on the torus, a

polygonal representation of M, and the type of P.












b





r


C








c


a 9 a


Figure 16

Figure 16. An example of a polyhedral map M, a polygonal representation P of M,
and the type of M.


a a


c d e f g a b c a b c a


a. b.

Figure 17. Two polygonal representations of a polyhedral map on the torus.


TI,











Lemma 3.1.1. Let M be a polyhedral map on a surface of Euler characteristic

X j 2, Tp the type of any polygonal representation of M, and v the number of distinct
vertex labels on Tp. If the vertices on Tp are labeled 1, ..., v and ni is the number of

occurrences of the label i on Tp, then n, + ... + n, = 2v + 2 2X. Furthermore, if

X = 1, then ni 3 for i = 1, ..., v.
Proof. First note that there cannot exist a vertex label that appears exactly

twice on Tp except in the case where M is a polyhedral map on the projective plane.
To see this, suppose B is a vertex label that appears exactly twice on Tp and let

A be another vertex label such that (A, B) is a directed edge on Tp and (A, B)'

its matching edge on Tp. If B = A, then there are no more vertex labels on Tp.

Hence the directed edge can be matched in exactly one way on Tp, and in this case

the surface is a projective plane. Next assume B $ A. If there are no more vertex

labels on Tp, then either the directed edges (A, B) and (B, A) can be concatenated

contradicting the fact that Tp is the type of a polygonal representation of M, or

the directed edge (B, B) cannot be matched on Tp, which is again a contradiction.

Hence, there must be another vertex with label C (possibly A) such that (B, C) is a

directed edge on Tp. Since the vertex label B appears exactly twice on Tp, there is

only one possibility for the matching edge (B, C)'. But the directed edges (A, B) and

(B, C) can be concatenated, which is a contradiction. Hence ni > 3 for i = 1 ..., v.

Next, consider the map M' with one face (the polygon Tp itself) obtained by
identifying matched directed edges on Tp and let e be the number of edges in M'.

Since the directed edges are matched in pairs on Tp, e 2 +.. It follows from

the Euler formula v e + f = X that

n, + ... + n, = 2v + 2 2X/. (3.1)









40


A
A B



A I A B OAI A A A


DA B
A



A
A B




B IV A A A




A B

A
A
A





A A A VI B




A B
A

Figure 18. The types of polygonal representations of polyhedral maps
on the projective plane, torus, and Klein bottle.











Theorem 3.1.2. Let M = (G, S) be a polyhedral map.

(1) If S is a projective plane, then M has a polygonal representation of type I in

Figure 18.
(2) If S is a torus, then M has a polygonal representation of type II or type III

in Figure 18.

(3) If S is a Klein bottle, then M has a polygonal representation of type IV, type

V, type VI, or type VII in Figure 18.
Proof. Consider the map M' with one face (the polygon P itself) obtained by

identifying matched directed edges on OP. Let v be the number of vertices and e the

number of edges on M'. Denote the vertex labels on OP by 1,2, ..., v. Further, let ni

denote the number of occurrences of the label i on OP. First consider the case where

M is a polyhedral map on the projective plane. By Lemma 3.1.1, v = 1, ni = 2; and

P is of type I. Next consider the case where M is a polyhedral map on the torus or

Klein bottle. Since X = 0 in this case, by Lemma 3.1.1, ni > 3 for i = 1,..., v and



n, + ... + n, = 2v + 2. (3.2)

Since equation (3.2) has no solutions for v > 2, v = 1, or v = 2. Consider the

following cases:
(1) v = 1 : In this case, there is exactly one vertex label on Tp and P must be of

type III in the case of the torus and of type V or type VI in the case of the Klein

bottle.
(2) v = 2 : In this case two vertex labels A and B appear exactly three times on Tp.

Furthermore, P must be of type II in the case of the torus and of type IV or type

VII in the case of the Klein bottle. M











A face F of a polygonal representation P is called separating if F n OP is discon-

nected. That is to say, the cycle OP revisits F. A polygonal representation without

separating faces is called non-separating, otherwise it is called separating. In the

example in Figure 16, the polygonal representation is non-separating, however both

polygonal representations shown in Figure 17 are separating. Specifically in Figure

17b, aP revisits the face labeled F. The existence of a non-separating polygonal rep-

resentation is a useful property of a polyhedral map. In the context of non-revisiting

paths, if a polyhedral map M has a non-separating polygonal representation P, then

any two vertices of M that lie in the interior of the polygon OP can be joined by

a non-revisiting path in M (see Proposition 3.1.2). Of course, given two vertices

of M, it is not always possible to find a polygonal representation P of M with the

property that the two vertices lie in the interior of the polygon OP for otherwise,

the non-revisiting path conjecture would be true for all polyhedral maps. And in

the context of non-revisiting cycles of a polyhedral map (this is discussed in Section

3.2.), the existence of non-separating polygonal representations enables us to give

simple proofs of results on non-revisiting cycles due to Barnette [6]. In addition,

it motivates the formulation of an interesting graph-colouring conjecture (discussed

in Section 3.3). However, not all polyhedral maps have a non-separating polygonal

representation. In fact, there is an infinite family of polyhedral maps that cannot

have any non-separating polygonal representations.


Proposition 3.1.1. For n > 7, if n =- 0,3,4,7(mod 12) and = (n3n4) ,
then there exists a triangulation of the orientable surface S, that is a polyhedral map

M, with the property that every polygonal representation of Mn is separating.

Proof. It is well known [11] that with 7 as above, the complete graph on n vertices

K, embeds on S.. Let Mn = (K,, S,) be a resulting map on S.,. If 1 < n < 4, then











-7 = 0 and the surface is the 2-sphere. If n = 5 or 6, then K, embeds on the torus.

However, the embedding is not a triangulation of the torus. So assume that n > 7.

If n = 0, 3, 4, 7(mod12), then (n-3)(n-4) is an integer and any embedding of K, on
12
S_ is, in fact, a triangulation of Sy. To see this, let v, e, and f be the number of

vertices, edges, and faces respectively, of M. Then v = n and e = n Hence
2
by the Euler equation for S_, f =le. Hence the embedding is a triangulation of S.

Since there are no multiple edges between vertices, the faces of the embedding meet

properly and the map Mn = (K, S.) is a polyhedral map.

Claim. For each n as above, if Mn has a non-separating polygonal representation

P then P,* is contained in M,* and has the following properties:

(1) The faces of Pn* are (n 1)-gons.

(2) Pn* has either two, three, or four faces.

(3) The graph of Pn* is a planar, spanning, 2-connected subgraph of the graph of

M.

Proof of Claim. Statement (1) is obvious. The vertices of Pn that lie in the interior

of the polygon 9Pn span a complete subgraph of Kn that is also contained in the

interior of the polygon OPn. If the number of vertices of Pn that lie in the interior of

the polygon OPn is greater than four, then by the previous statement, the graph of Pn

would be non-planar, which is a contradiction since P, is a planar map. Consequently,

Pn* can have at most four faces. If Pn* has no faces, then P. must be separating,

which is a contradiction. If Pn* has exactly one face, then Pn must have faces that

meet improperly, which is also a contradiction since P, is a polyhedral map. To see
statement (3), note that the graph of Ps* is planar, has all the vertices of M,,, and

is 2-connected because Pn was assumed to be non-separating.
Hence, for each n > 7, if P,,* has exactly two faces, then the graph of Ps* has

2n 6 vertices. On the other hand, if P* has exactly three faces, then the graph of











Pn* has 3n 7 or 3n 8 vertices and if P,* has exactly four faces, then the graph

of P has 4n 10,4n 11, or 4n 12 vertices. Now, by the Euler formula, M* has
v* n(n-1) n + 2 (n-3)(n-4) vertices. Note that as n increases, the number of
2 6

vertices of the graph of Pn* grows linearly while v* grows quadratically. In fact, for

n > 11, v* is greater than each of the numbers 2n 6, 3n 7, 3n 8, 4n 10, 4n 11

and 4n 12. Thus if n > 11, the graph of Pn* cannot possibly span the graph of

Mn and we need only consider the case where n = 7. Let M7 be the polyhedral map

corresponding to the polygonal representation shown in Figure 17a. By statements

(1), (2) and (3) above, P* must be a map on a closed disc with 14 vertices and 3

hexagonal faces. Hence the only possibilities for P7* are as shown in Figure 19.












Figure 19. The two possibilities for a 2-connected, planar
map with 14 vertices and 3 hexagonal faces.

However, it is easily checked that these planar maps are not contained in M*. n


In Section 2.2, the polyhedral maps M1 and M2 were counterexamples to the

non-revisiting path conjecture for the surfaces S2 and N4, respectively. Figure 20a

shows a polygonal representation of M1 while Figure 20b gives a polygonal represen-

tation for M2. Observe that both polygonal representations are separating, however,

it is easy to construct similar counterexamples that have non-separating polygonal

representations.


















1 1
C DD C





2A B 4 2 ... A B

D
C "



YY


3 C 3



a b



Figure 20. Polygonal representations for the polyhedral maps in
Figures 12 and 14.

Proposition 3.1.2. Let M be a polyhedral map that has a non-separating polyg-

onal representation P. Then any two vertices of M that lie in the interior of the

polygon P can be joined by a non-revisiting path in M.
Proof : Let u and v be vertices of M that lie in the interior of the polygon P.

Since P is non-separating, by Lemma 2.1.2, there is a path F joining u and v in M

that is also contained in the interior of the polygon P. Clearly, F can have only











planar revisits. Hence by Lemma 2.1.1, there is a non-revisiting path joining u and

v in M.




3.2 Polygonal Representation and Non-Revisiting Cycles




It is known that any polyhedral map on the projective plane, torus or Klein bottle

has a non-planar, non-revisiting cycle. Barnette's proofs [6] of these results are not

trivial and involve some details. In this section, we give a unified, elementary proof

of these results in the case where the polyhedral map has a non-separating polygonal

representation. This is done by considering the cycles of M that lie on the boundary

of a non-separating polygonal representation of M. Such cycles are non-planar by

Lemma 3.2.1 below. And by techniques that are similar to those used in Chapter

2, it is shown that if a cycle of M contained in OP revisits a face, then it can be

modified to a cycle that is non-revisiting and is also contained in OP. In the next

section, a graph-colouring conjecture is proposed and it is shown that the conjecture

is true for all planar graphs. Consequently, an alternate proof of the above-stated

result on non-revisiting cycles is given.

Lemma 3.2.1. Let M = (G, S) be a polyhedral map on a surface and P be a

polygonal representation of M. If C is a cycle of M that is contained in OP, then it

must be a non-planar cycle in M.

Proof. The proof is by contradiction; so assume that C bounds a cell A in M. Let

f be a face of M that is not contained in A and that has an edge e in common with C.

Such a face exists; otherwise A would contain all the faces in M which is impossible.

Let e' be the matching edge for e on OP and let f' be the face of P that contains











e'. Since f does not belong to A, f' must belong to A. Also, since P is connected,
so is its dual P*. Also, by definition, P* n OP = 0. Hence, with vj and vf, as the

vertices of P* corresponding to the faces f and f' of P, respectively, there is a path

v1, v1,, ..., vfk, v1, from vf to v1, in P* that is contained in the interior of the polygon

OP. Hence, there is a sequence of faces fi, i = 1, ..., k of P corresponding to the ver-

tices vfi of P* such that f n fl, f, n f2, ..., fk-I n fk, fk n f', are all edges of P that are

contained in the interior of the polygon OP. But the edges of C are all on OP. Hence

the interior edges f n fl, fl n f2,..., fk- I n fk, fk n f', are all in the cell A. This implies

that the face f is also in A, which is a contradiction to the choice of f as a face of

M not in A. Hence C does not bound a cell on the surface and must be non-planar. m


Corollary 3.2.1. Every polyhedral map on a surface (except on the sphere) has

a non-planar cycle.
Proof. Let M be a polyhedral map and let P be a polygonal representation of

M. By the definition of a polygonal representation, every vertex on OP appears at

least twice on 9P. Hence there is at least one cycle that is contained in OP that

is obtained by traveling along OP between two consecutive vertices both labeled A

that have the property that there is no other pair of matched vertices that appear

between the A's. By Lemma 3.2.1, such a cycle must be non-planar. 0



Theorem 3.2.1 Let M be a polyhedral map on the projective plane, torus, or

Klein bottle. If M has a non-separating polygonal representation, then M has a

non-planar, non-revisiting cycle.











Proof. First consider the case where M is a polyhedral map on the projective

plane. By Theorem 3.1.1, there is a polygonal representation P of M that is of type

I. Let C be the cycle (A, A) along &P as shown in Figure 21.












A T A






Figure 21. Non-planar, non-revisiting cycles on the projective plane.






The only possibility for a revisit of C to a face F is if F n OP is disconnected.

But this contradicts the assumption that P is non-separating. Hence C must be

non-revisiting. And by Lemma 3.2.1, C is also non-planar.

Next, consider the case where M is a polyhedral map on the torus. By Theorem

3.1.1, there is a polygonal representation P of M that is of type II or type III. First

consider the case where P is of type II. Let C = (A, A) be the bold-faced cycle shown

in Figure 22











A s B





B* A





A s B


a.

Figure 22. Non-planar, non-revisiting cycles on the torus.

If C has a revisit to a face F, then it can be easily checked that F must contain

both A and B for otherwise OP would have to revisit F which contradicts the as-

sumption that P is non-separating. Up to symmetry, F must be as shown in Figure

22. Replace C by the cycle C1 = (B, B) as shown. If C, revisits a face F1, then it

can be checked that F must also contain the vertices labelled A and B. But this

means that F and F meet improperly at A and B which is a contradiction. Hence

C, must be non-revisiting. Next, consider the case where P is of type III and let C

be the cycle (A, A) along OP as shown in Figure 23. By the same argument as in the

case of the projective plane, it is easy to see that C is non-planar and non-revisiting.

Finally, consider the case where M is a polyhedral map on the Klein bottle and

let P be a polygonal representation of M that is of type IV. Let C = (A, A) be

the cycle shown in Figure 24. If C is non-revisiting, we are done, so assume that C

revisits a face F. As in the case of the torus, it can be checked that F must contain

both A and B. There are two possibilities for F as shown in Figure 24.


























Figure 23. Non-planar, non-revisiting cycles on the torus.


A t B


A B


Figure 24. Non-planar, non-revisiting cycles on the Klein bottle.

In both cases, replace C by the cycle C1 = (B, B) as shown. Now C, must be

non-planar and non-revisiting.
If P is of type V, then by the same argument as for the projective plane the cycle

C shown in Figure 25 is non-planar and non-revisiting. If P is of type VI or type

VII, then the cycles shown in Figures 26a and 26b, respectively can be easily checked

to be non-planar and non-revisiting.









51


A




AC
A type V

A






A

Figure 25. Non-planar, non-revisiting cycles on the Klein bottle.









A
A




A A A B








A A B


a. b.

Figure 26. Non-planar, non-revisiting cycles on the Klein bottle.











Thus in all cases M has a non-planar, non-revisiting cycle that is contained in
OP. U




3.3 A Graph-Coloring Problem and Non-Revisiting Cycles




A path P in a graph G is said to be a chord of a cycle C in G if P is a path joining

vertices x and y of C such that P n c = {x, y}. For our purposes, an edge-coloring

of G is a coloring of the edges of G in which every edge can be colored with many

different colors. Given such an edge-coloring of G, a subgraph H of G is said to be

monochromatic if there is a color C such that every edge of H is colored with C1.

Similarly, H is said to be dichromatic if there are colors C, and C2 such that every

edge of H is colored with C, or C2.



Conjecture. Every edge-colored finite graph G with no mono or dichromatic cy-

cles contains a cycle with no monochromatic chord.



The above conjecture is motivated by the problem of the existence of non-planar,

non-revisiting cycles in a polyhedral map. If every edge is coloured using exactly

two colors, then the validity of the coloring conjecture implies that every polyhedral

map that has a non-separating polygonal representation, in fact, has a non-planar,

non-revisiting cycle. The proof of this result follows.











Theorem 3.3.1. If the conjecture is true in the case where each edge is colored

with exactly two colors, then every polyhedral map with a non-separating polygonal

representation contains a non-planar, non-revisiting cycle.
Proof. Let P be a non-separating polygonal representation of a polyhedral map

M on a surface. Let {Fi}, i =1, ..., k, be the collection of faces of M that have at

least one edge in common with OP. Since P has no separating faces, for i =1, ...,

PA = Fi nOP is a path in OP. For i = 1, ..., k, color the edges of the path Pi using

a distinct color Ci. Since the edges on OP are matched in pairs, every edge in M

that lies on OP is colored using exactly two colors. Now consider OP and identify

the matched edges on OP. The result is a graph G, where each edge is colored using

exactly two colors. Note that the cycles in G are the cycles of M contained in OP.

Recall, by Lemma 3.2.1, that the cycles that are contained in OP are non-planar.

Furthermore, there is a monochromatic cycle in G if and only if, for some i, two

vertices of Pi are identified. This in turn implies the face F of M is not simply-

connected which is impossible. There is a dichromatic cycle in G if and only if there

are faces Fi and Fj of M that meet improperly which is also not allowed. Finally, a

cycle in G has a monochromatic chord using a color Ci if and only if the corresponding

cycle on OP revisits the face Fi of M. Hence, if the conjecture is true, then there

must be a cycle of M contained in OP that has no monochromatic chord and hence

must be non-revisiting. 0

Example 3.3.1. The conjecture is true for all graphs that contain a triangle.

Proof. Let G be a graph and T a triangle of G. The proof is by contradiction;

so assume that G has an edge-coloring with no mono or dichromatic cycles such that
every cycle of G has a monochromatic chord. In particular, T has a monochromatic

chord P. Since there are no multiple edges between vertices, P has length at least











two. Let v, and v2 be such that P n T = {V1, v2} and e = V1V2 the edge in T. Then

the cycle P[V1, v2] U {e} is a mono or dichromatic cycle, which is a contradiction. ,

Example 3.3.2. The conjecture is true for K3,3.

Proof. The proof is by contradiction; so assume that there is an edge-coloring of

K3,3 with no mono or dichromatic cycles such that every cycle has a monochromatic

chord. Let the vertices of K3,3 be labeled as shown in Figure 27.
1


c A



3 *2


B
Figure 27. The graph-coloring conjecture is true for K3,3.

It will be shown that there must be a 4-cycle with no monochromatic chord. It

is easy to see that a monochromatic chord of any 4-cycle must have length at least

two. If a monochromatic chord of a 4-cycle has length greater than two, then the

endpoints of the chord are adjacent to each other in K3,3. This is a contradiction since

in this case, the monochromatic chord together with the edge joining the endpoints

of the chord form a mono or dichromatic cycle. Hence a monochromatic chord of

a 4-cycle must have length exactly two. First, consider the 4-cycle C1 = 1A3C1

and let P be a monochromatic chord of C1. Up to symmetry, P = C2A. Next,

the cycle C2 = 1A2C1 must have a monochromatic chord P2. The vertices labeled

A and C cannot be the endpoints of P2 for then, the paths P and P2 form a mono

or dichromatic cycle. Hence P2 = 1B2. Let C3 = 1A2B1 and P3 a monochromatic

chord of C3. By a similar argument as the one given for P2, P3 = A3B. If P4











is a monochromatic chord for the cycle C4 = 2B3A2, then P4 = 2C3. Likewise, if

C5 = 2B3C2, then the monochromatic chord P5 of C5 is B1C; and with C6 = 1B3C1,

the monochromatic chord P6 of C6 must be 3A1. Finally, consider the 4-cycle C7 =

A3C2A and let P7 be a monochromatic chord of C7. There are two possibilities for

P7. If P7 = 3B2, then P4 and P7 form a mono or dichromatic cycle. On the other

hand, if P7 = CiA, then P and P7 form a mono or dichromatic cycle. In either case,

its a contradiction. Hence C7 cannot have a monochromatic chord.


Theorem 3.3.2. The conjecture is true for all planar graphs.
Proof. The proof is by contradiction; so assume that there is a planar graph G

for which the conjecture is false. In other words, for some edge-coloring of G with no

mono or dichromatic cycles, every cycle of G has a monochromatic chord. Consider

an embedding of G in the plane in which edges cross each other only at vertices of G.

Let WoC be the boundary of the unbounded face of G and Co the closed region interior

to 6Co. Let P be any monochromatic chord of Co. Then P must be contained

in Co and it separates Co into components C1 and B1 such that C, n B1 = P and

C, U B1 = Co. Now consider the cycle 0C1 that bounds the component C, and let P2

be any monochromatic chord for 0C1. If P2 leaves the component C1 then it must

enter the component B1 by crossing the monochromatic chord P1 and the only way

for P2 to re-enter C1 is for it to cross P again. But this yields a dichromatic cycle,

which is not allowed. Hence the only way 8C, can have a monochromatic chord P2

is for P2 to be completely contained in C1. Now P2 separates C1 into components

C2 and B2 such that C2 n B2 = P2 and C2 U B2 = P1. For every integer k > 1, we

claim that at the kth step, if Pk+l is any monochromatic chord for the cycle 8Ck,

then the component Ck is divided into components Ck+1 and Bk+1 with the following

properties:











(1) Ck+1 U Bk+1 = Ck.

(2) Cki nBk+ = Pk+,.

Conditions (1) and (2) above are equivalent to the statement that Pi+1 is contained

in Ck. The proof of the above claim is by induction on k. It was shown above that

the claim is true for k = 1. Assume that the claim is true for each i < k. If Pk+i, a

monochromatic chord for W0k, is contained in Ck, we are done, so assume that Pk+1

leaves Ck. However, if Pk+l does not leave Ck-i, then the only way it can return to

Ck is by crossing the monochromatic chord Pk twice giving a dichromatic cycle in G,

which is not allowed. On the other hand, if Pk+i leaves both Ck and Ck-1, then it

eventually has to return to Ck. But by condition (1) for the component Ck-1, this

means that it also returns to Ck-1. Thus, there is a subpath Qk+1 of Pk+i that is

a monochromatic chord for the cycle aCk-1 and that returns to Ck-1 after leaving.

This is a contradiction to the induction hypothesis. Hence Pk+1 cannot leave Ck,

proving the claim.

It follows that G would have to be an infinite graph in order that every cycle in G

have a monochromatic chord, which is a contradiction. Hence there can be no planar

graph that can be a counter-example to the conjecture and the theorem is true for

all planar graphs. N


As an application of Theorem 3.3.1, we give another simple proof of Theorem
3.2.1


An alternate proof of Theorem 3.2.1 In the case of each surface, let P be

a non-separating polygonal representation of M and let GT and G be the graphs

obtained by identifying the edges on Tp and 8P, respectively. Since Tp is obtained

from aP by performing concatenations along &P, G can be obtained from GT by











inserting vertices of G that are not in GT along the interior of each edge of GT. Thus

GT and G are homeomorphic. If M is a polyhedral map on the projective plane, then

by Theorem 3.1.2, P is of type I and consequently, GT is isomorphic to the graph in

Figure 28a. By a similar argument, if M is a polyhedral map on the torus, then G is

homeomorphic to the graph in Figure 28b, or Figure 28c. and if M is a polyhedral

map on the Klein bottle, then G is homeomorphic to the graph in Figure 28b, Figure

28c, or Figure 28d.



0 D

A

a. b. C. d.

Figure 28. The boundary graphs of polyhedral maps on the
projective plane, torus, and Klein bottle.

In all cases, G is planar and by Theorem 3.3.2, the conjecture is true. Hence by

Theorem 3.3.1, in each case, there is a non-planar, non-revisiting cycle in M. 0




So far, two elementary proofs of Theorem 3.2.1 have been provided. However, the

scope of Theorems 3.3.1 and 3.3.2 are greater than merely giving proofs for already

known results on non-planar, non-revisiting cycles on the projective plane, torus,

and Klein bottle. In this regard, the following result extends Barnette's result on

non-revisiting cycles on the three surfaces mentioned above to a class of polyhedral

maps on the surfaces N3 or S2.











Theorem 3.3.3. Every polyhedral map on N3 or S2 that has a non-separating

polygonal representation contains a non-planar, non-revisiting cycle.

Proof. Let M be a polyhedral map on N3 and assume that M has a non-

separating polygonal representation P. Let the vertices of Tp be labeled 1, ..., v and

for i = 1, ..., v, let ni be the number of occurrences of the label i on Tp. Since the

Euler characteristic in this case is -1, by Lemma 3.1.1,

n, + ... + n, = 2v + 4. (3.3)

where ni > 3 for i = 1, ..., v. It is easily checked that the above equation has no

solutions for v > 5. That is to say, there are at most four different vertex labels on

Tp. Hence the graph GT, obtained by identifying matched directed edges on Tp can

have at most four vertices and consequently, must be planar. By the same argument

given in the alternate proof to Theorem 3.2.1, the graph G obtained by identifying

matched edges on OP must be homeomorphic to GT and consequently, must also be

planar. By Theorem 3.3.2, the conjecture is true for the graph G. Hence by Theorem

3.3.1, M has a non-planar, non-revisiting cycle that is contained in OP.

Next, consider the case where M is a polyhedral map on S2 and let P be a non-

separating polygonal representation of M. Since the Euler characteristic in this case

is -2, by Lemma 3.1.1,

n, +... + n, = 2v + 6. (3.4)

where ni > 3 for i = 1,...,v. It is easily checked that the above equation can have

solutions only if v < 6. If e is the number of distinct directed edges on Tp, then e < 9.

Let G be the graph obtained by identifying matched edges on Tp. Then G has at

most 6 vertices and at most 9 edges. Since K5 has 10 edges, G is either planar, or is

isomorphic to K3,3. In either case, the coloring conjecture for two colors is true, and








59


by the same argument given for the surface N3, there is a non-planar, non-revisiting

cycle in M that is contained in OP. M












CHAPTER 4
CONCLUSION




The non-revisiting path conjecture is now settled for all polyhedral maps except

those that are homeomorphic to N3. It is conceivable that the conjecture is true

for polyhedral maps on surfaces homeomorphic to N3, however, at this juncture, no

proof is known. Considering the complexity of the proofs of the validity of the non-
revisiting path conjecture for the torus and Klein bottle, a brute force method might

prove to be tedious in the case of the surface N3.

The problem of the existence of non-planar, non-revisiting cycles in a polyhedral

map is wide open The only surfaces for which such cycles are known to exist are the

projective plane, torus, Klein bottle, and for a class of polyhedral maps on the surfaces

homeomorphic to N3 and S2 (see Theorem 3.3.3). If every polyhedral map contains a

non-planar, non-revisiting cycle, one could potentially obtain decomposition theorems

for surfaces other than the torus and Klein bottle.

Even though the graph coloring conjecture proposed in Section 3.3 was motivated

by the existence of non-planar, non-separating cycles in a polyhedral map, the conjec-

ture is certainly interesting on its own merit. Apart from planar graphs, graphs with

a triangle, IK3,3, and some very specific graphs (not included in this dissertation), the

coloring conjecture remains unsettled.

In conclusion, this research has raised at least three different questions that remain

unsolved and would make for some interesting work in the future.











REFERENCES


[1] Adler, I. (1974). Lower Bounds for Maximum Diameters of Polytopes, Pivoting
and Extensions. Math. Programming Study. 1, 11-19.
[2] Balinski, M. L. (1961). On the Graph Structure of Convex Polytopes in n-Space.
Pacific Journal Of Math. 11, 431-434.
[3] Barnette, D. W. (1969). W, Paths on 3-Polytopes. J. Combinatorial Theory. 7,
62-70.
[4] Barnette, D. W. (1973). Graph Theorems for Manifolds. Israel Journal Of Math.
16, 62-72.
[5] Barnette, D. W. (1986). W Paths in the Projective Plane. Discrete Math. 62,
127-131.
[6] Barnette, D. W. (1988). Decomposition Theorems for the Torus, Projective
Plane and Klein Bottle. Discrete Math. 70, 1-16.
[7] Barnette, D. W. (1990). W, Paths on the Torus. Discrete Comp. Geom. 5, 603-
608.
[8] Barnette, D. W. (1993). A 2-manifold of Genus 8 Without the We-Property.
Geometriae Dedicata. 46, 211-214.
[9] Dantzig, G. B. (1963). Linear Programming and Extensions. Princeton Univer-
sity Press, Princeton, N.J.
[10] Engelhardt, E. (1988). Some Problems on Paths in Graphs. Ph. D. thesis. Uni-
versity of Washington, Seattle, Wa.
[11] Gross, J. L. and Tucker, T. W. (1987). Topological Graph Theory. Wiley Inter-
science Series in Discrete Mathematics and Optimization.
[12] Kalai G. and Kleitman D. (1992). A Quasi-Polynomial Bound for the Diameter
of Graphs of Polyhedra. Bull. Amer. Math. Soc. 226(2), 315-316.
[13] Klee, V. (1965). Problem 19, Colloquium On Convexity. (Copenhagen).
[14] Klee, V. (1965). Paths on Polyhedra I. J. Soc. Indust. Appl. Math. 13, 946-956.
[15] Klee, V. (1966). Paths on Polyhedra II. Pacific Journal Of Math. 17, 249-262.
[16] Klee, V. and Walkup D. (1967). The d-Step Conjecture for Polyhedra of Dimen-
sion, d < 6. Acta Math. 133, 53-78.
[17] Klee, V. and Kleindschmidt, P. (1987). The d-Step Conjecture and its Relatives.
Mathematics Of Operations Research. 12(4), 718-755.
[18] Larman, D. G. (1974). Paths on Polytopes. Proc. Lond. Math. Soc. 3, 161- 178.

61











REFERENCES


[19] Mani, P. and Walkup, D. (1980). A 3-Sphere Counterexample to the W, Path
Conjecture. Math. Oper. Res. 5(4), 595-598.
[20] Steinitz, E. and Rademacher, H. (1934). Vorlesungen Uber Die Theorie Der
Polyeder. Springer, Berlin.
[21] Tutte, W. T. (1984). Graph Theory. Addison-Wesley Publishing Company.











BIOGRAPHICAL SKETCH



Hari Pulapaka was born in Bombay, India, on March 19, 1966. Upon receiving a

bachelor's degree in mathematics from St. Xavier's College, University of Bombay,

he arrived in the U.S. in 1987 to attend graduate school. In 1989, he received an M.S.

in mathematics from George Mason University, Fairfax, VA, under the supervision

of Dr. James Lawrence. And in 1995, he received a Ph.D. in mathematics from the

University of Florida, Gainesville, FL, under the supervision of Dr. Andrew Vince.










I certify that I have read this study and that in my opinion it conforms to accept-
able standards of scholarly presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philo y.


professorr of Mathematics

I certify that I have read this study and that in my opinion it conforms to accept-
able standards of scholarly presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.


Krishnaswami Alladi
Professor of Mathematics

I certify that I have read this study and that in my opinion it conforms to accept-
able standards of scholarly presentation and is fui adequ in scope and quality,
as a dissertation for the degree of Doctor of P osoph

Timothy Dwe ,/
Assistant Professor d Computer and
Information Sciences

I certify that I have read this study and that in my opinion it conforms to accept-
able standards of scholarly presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philophy.


Bernard Mair
Associate Professor of Mathematics

I certify that I have read this study and that in my opinion it conforms to accept-
able standards of scholarly presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philos w


Neil Lt hite
Professor of Mathematics

This dissertation was submitted to the Graduate Faculty of the Department of
Mathematics in the College of Liberal Arts and Sciences and to the Graduate School
and was accepted as partial fulfillment of the requirements for the degree of Doctor
of Philosophy.

A ugust 1995 D ean, G raduate School
Dean, Graduate School




Full Text
51
A
Figure 25. Non-planar, non-revisiting cycles on the Klein bottle.
a.
b.
Figure 26. Non-planar, non-revisiting cycles on the Klein bottle.


NOTATION
G
V(G)
E(G)
S
X
dS
S9
Nk
M
M*
P
TP
A finite graph.
The vertex set of G.
The edge set of G.
A surface.
The Euler Characteristic of a surface.
The boundary of the surface S.
A surface homeomorphic to the connected sum of g tori.
A surface homeomorphic to the connected sum of k projective planes.
A polyhedral map.
The dual poyhedral map.
A polygonal representation of a polyhedral map.
The type of a polygonal representation P.
vi


The second problem concerns the existence of non-planar, non-revisiting cycles in
a polyhedral map on a surface. By results due to Barnette, it is known that every
polyhedral map on the projective plane, torus, and Klein bottle contains a non-
planar, non-revisiting cycle. In this regard, the notion of a polygonal representation
of a polyhedral map is introduced. This is analogous to the notion of representing
a surface as a polygon in the plane with the directed sides of the polygon matched
in pairs, except in this case, the representation preserves the combinatorial structure
of the underlying graph of the polyhedral map. Some properties of polygonal rep
resentations are proved. As an application, an elementary, unified proof of results
due to Barnette involving the existence of non-planar, non-revisiting cycles on the
projective plane, torus, and Klein bottle is given.
The third problem is a graph-coloring conjecture that is shown to be true for
all planar graphs and /T3)3. As an application, it is shown that any polyhedral
map on a surface homeomorphic to the connected sum of three projective planes, or
the connected sum of two tori, that has a non-separating polygonal representation,
contains a non-planar, non-revisiting cycle. This extends Barnettes result stated in
the second problem.
Vlll


55
is a monochromatic chord for the cycle C4 = 253A2, then 54 = 2(73. Likewise, if
C5 = 253(72, then the monochromatic chord P5 of C5 is BlC; and with C& = 153(71,
the monochromatic chord 56 of C& must be 3A1. Finally, consider the 4-cycle C7 =
A3C2A and let P7 be a monochromatic chord of C7. There are two possibilities for
P7. If P7 = 352, then 54 and P7 form a mono or dichromatic cycle. On the other
hand, if P7 = CIA, then Pi and P7 form a mono or dichromatic cycle. In either case,
its a contradiction. Hence C7 cannot have a monochromatic chord.
Theorem 3.3.2. The conjecture is true for all planar graphs.
Proof. The proof is by contradiction; so assume that there is a planar graph G
for which the conjecture is false. In other words, for some edge-coloring of G with no
mono or dichromatic cycles, every cycle of G has a monochromatic chord. Consider
an embedding of G in the plane in which edges cross each other only at vertices of G.
Let dCo be the boundary of the unbounded face of G and Co the closed region interior
to dCo. Let Pi be any monochromatic chord of dCo. Then Pi must be contained
in Co and it separates Co into components Ci and 5i such that Ci D 5i = 5X and
Ci U 5i = Co- Now consider the cycle dCi that bounds the component Ci and let P2
be any monochromatic chord for dCi. If 52 leaves the component Ci then it must
enter the component 5X by crossing the monochromatic chord 5X and the only way
for 52 to re-enter Ci is for it to cross 5X again. But this yields a dichromatic cycle,
which is not allowed. Hence the only way dCi can have a monochromatic chord 52
is for P2 to be completely contained in Cj. Now 52 separates C\ into components
C2 and 52 such that <72 D 52 = 52 and (72 U 52 = 5X. For every integer k > 1, we
claim that at the kth step, if Pk+i is any monochromatic chord for the cycle dCk,
then the component Ck is divided into components Ck+i and 5+1 with the following
properties:


17
interior of H. Such a path can have only planar revisits and by Lemma 2.1.1, we are
done.
Next consider the case where M is a polyhedral map on the torus. By Lemma
2.1.4, M has a non-revisiting cycle C such that cutting M along C yields an annulus
A. Let Ci and (72 be the bounding cycles of A corresponding to C and without loss
of generality, consider the following three cases :
1. u and v lie on C. In this case, either of the two paths along C from u to v is
non-revisiting (since C is non-revisiting).
2. u lies on C and v does not lie on C. As in the case of the projective plane, there
is a path T joining u and v that meets dA at only u. Without loss of generality,
assume that 1 meets C\ at only u and avoids C*2. If all of Ts revisits are planar, we
are done. So assume that T has a non-planar revisit (s,t) to a face F with the vertex
s closer to v along T than the vertex t is to v. Among all non-planar revisits of T
choose F so that s is nearest to v along T. Note that F cannot meet both C\ and C2
in A for this would mean that the cycle C revisits F which is a contradiction to the
assumption that C is a non-revisiting cycle. Consider the following two cases :
i. F contains u. Up to symmetry, there are two possibilities for F (depending on
whether F meets C\ or C2) as shown in Figure 4.


31
Paste the polygons together by identiying the edges with the same labels. It can
be checked that the result is a surface S without boundary with the map Ml given
by the union of the 16 polygons. Next, it is shown that S is orientable. This is done
as follows:
First note that each face has two possible directions for its boundary walk. Assign
an orientation to each face by choosing one of these two directions. If every face
can be assigned an orientation in such a way that adjacent regions induce opposite
directions on every common edge, the surface S is orientable. Such an orientation
for the faces F\,..., Fie is shown in Figure 13.
4 A 1 B 3 A
D
> 1
O
<
O
B 4
O
O '
O
X
O
O

0
L
3
C
1 C
D
C
L D
3 D
2 A
4
r i
\o
G\
/
O/
/O
\o
0\

O/
/o
V- c
3
4
B
B 2
c
Figure 13. An orientation on the faces
in Figure 12 that shows that
the surface is orientable.
Observe that Mi has 10 vertices, 28 edges, and 16 faces. Hence the Euler Char
acteristic of S is 2. And since S is orientable, it must be homeomorphic to S2. It is


36
in pairs and the surface may be obtained by identifying the matched directed sides
of the polygon. Analogously, if M (G, S) is a polyhedral map, then a polygonal
representation of M is a representation of M as a polygon in the plane that preserves
the combinatorial structure of G. Thus the sides of the polygon are in fact, edges in
G. This notion is made more precise below.
A polygonal map P is defined as a polyhedral map on a closed disc such that:
(1) The vertices of dP are labeled, and every label appears at least twice on dP.
(2) The edges of dP are directed and there is a matching on this set of directed edges
of dP that matches each directed edge labeled (A, B) with another directed edge
labeled (A, B) with the same labels.
If M is a polyhedral map on a surface, then a polygonal map P is called a polygonal
representation of M if
(1) M is obtained from P by identifying matched edges on dP and,
(2) after the identifications, each vertex label appears exactly once in M.
Note that, in general, a polyhedral map may have several polygonal representa
tions. Figure 17 shows two polygonal representations of a polyhedral map (whose
underlying graph is K7) on the torus. Also if P is a polygonal representation of M,
then there can be no matched edges on dP as in Figure 15 below. Otherwise label
A either appears only once on dP, contradicting statement (1) in the definition of a
polygonal map, or label A appears more than once in M, contradicting statement (2)
in the definition of a polygonal representation. A polyhedral map and a polygonal
representation of the polyhedral map are shown in Figure 16.


ACKNOWLEDGEMENTS
First and foremost, I would like to thank Professor Vince for his invaluable help
and advice during the preparation of this dissertation. He has spent countless hours
helping me understand the subtleties of my work. Sifting through some of the first
drafts must have been quite painful, and for this, I remain deeply appreciative of him.
Also, I would like to thank Professors Alladi, Davis, Mair, and White for taking the
time to serve on my supervisory committee.
Finally, I would like to dedicate this work to my parents and Cynthia. With
out their moral support, this endeavor would be impossible and more importantly,
meaningless.
11


45
a
b
Figure 20. Polygonal representations for the polyhedral maps in
Figures 12 and 14.
Proposition 3.1.2. Let M be a polyhedral map that has a non-separating polyg
onal representation P. Then any two vertices of M that lie in the interior of the
polygon P can be joined by a non-revisiting path in M.
Proof : Let u and v be vertices of M that lie in the interior of the polygon P.
Since P is non-separating, by Lemma 2.1.2, there is a path T joining u and v in M
that is also contained in the interior of the polygon P. Clearly, T can have only


48
Proof. First consider the case where M is a polyhedral map on the projective
plane. By Theorem 3.1.1, there is a polygonal representation P of M that is of type
I. Let C be the cycle (A, A) along dP as shown in Figure 21.
Figure 21. Non-planar, non-revisiting cycles on the projective plane.
The only possibility for a revisit of C to a face F is if F D dP is disconnected.
But this contradicts the assumption that P is non-separating. Hence C must be
non-revisiting. And by Lemma 3.2.1, C is also non-planar.
Next, consider the case where M is a polyhedral map on the torus. By Theorem
3.1.1, there is a polygonal representation P of M that is of type II or type III. First
consider the case where P is of type II. Let C = (A, A) be the bold-faced cycle shown
in Figure 22


42
A face F of a polygonal representation P is called separating if F fl dP is discon
nected. That is to say, the cycle dP revisits F. A polygonal representation without
separating faces is called non-separating, otherwise it is called separating. In the
example in Figure 16, the polygonal representation is non-separating, however both
polygonal representations shown in Figure 17 are separating. Specifically in Figure
17b, dP revisits the face labeled F. The existence of a non-separating polygonal rep
resentation is a useful property of a polyhedral map. In the context of non-revisiting
paths, if a polyhedral map M has a non-separating polygonal representation P, then
any two vertices of M that lie in the interior of the polygon dP can be joined by
a non-revisiting path in M (see Proposition 3.1.2). Of course, given two vertices
of M, it is not always possible to find a polygonal representation P of M with the
property that the two vertices lie in the interior of the polygon dP for otherwise,
the non-revisiting path conjecture would be true for all polyhedral maps. And in
the context of non-revisiting cycles of a polyhedral map (this is discussed in Section
3.2.), the existence of non-separating polygonal representations enables us to give
simple proofs of results on non-revisiting cycles due to Barnette [6]. In addition,
it motivates the formulation of an interesting graph-colouring conjecture (discussed
in Section 3.3). However, not all polyhedral maps have a non-separating polygonal
representation. In fact, there is an infinite family of polyhedral maps that cannot
have any non-separating polygonal representations.
Proposition 3.1.1. For n > 7, if n = 0,3,4, 7{mod 12) and 7 [ (n~3)(n-4) j ^
then there exists a triangulation of the orientable surface Sf that is a polyhedral map
Mn with the property that every polygonal representation of Mn is separating.
Proof. It is well known [11] that with 7 as above, the complete graph on n vertices
Kn embeds on S-,. Let Mn (Kn, S-,) be a resulting map on S-,. If 1 < n < 4, then


REFERENCES
[19] Mani, P. and Walkup, D. (1980). A 3-Sphere Counterexample to the Wv Path
Conjecture. Math. Oper. Res. 5(4), 595-598.
[20] Steinitz, E. and Rademacher, H. (1934). Vorlesungen Uber Die Theorie Der
Polyeder. Springer, Berlin.
[21] Tutte, W. T. (1984). Graph Theory. Addison-Wesley Publishing Company.
62


30
of course contradicts Engelhardts result for the surface ST Since the non-revisiting
path conjecture is already known to be true for the 2sphere, projective plane, torus,
and Klein bottle, the only surface for which the non-revisiting path conjecture is still
open is AT
The Counterexamples. The polyhedral maps that constitute the counterexam
ples for the surfaces mentioned above will be described in terms of the polygons that
form the faces of the polyhedral map. Thus, the vertices and edges of the polyhedral
map are those of the polygons and the surface is obtained by glueing the polygons
together along the edges with the same labels.
First consider the orientable case and let Fi,...,Fi6 be the polygons with the
vertex-labelling shown in Figure 12.
3
Figure 12. The faces that constitute a counter-example to the
non-revisiting path conjecture for polyhedral maps on ST


3
on a surface without boundary, then the dual map of M, denoted by M* is defined
as follows: for each face / of M, place a vertex /* in its interior. Then, for each edge
e in G, draw an edge e* between the vertices just placed in the interiors of the faces
containing the edge e. The resulting graph with vertices f* and edges e* is called the
dual graph of G, denoted by (?*, and the resulting map (G*,S) is the dual map of
M. On the other hand, if S has boundary dS, then define the dual map as follows:
for each face / of M, place a vertex f* in its interior. Then, for each edge e in G not
belonging to dS, draw an edge e* between the vertices just placed in the interiors of
the faces containing the edge e. The resulting graph G* is called the dual graph of G
and the resulting map is called the dual map of M. An important property satisfied
by 3-connected graphs embedded on the 2-sphere is that any two faces intersect on a
single edge, a single vertex or not at all. Faces that meet in this way are said to meet
properly. If all the faces are simply-connected and all faces meet properly, then the
map M is called a polyhedral map on the surface. A consequence of all faces meeting
properly is that every vertex of a polyhedral map has degree at least three. By a
result of Barnette [4], every polyhedral map is 3-connected, generalizing Steinitzs
Theorem.
1.2 Some History and Motivation
The feasible region of any non-empty linear programming problem is a polyhedron,
and conversely, given a polyhedron P, it is always possible to construct a linear
program with P as its feasible region. Edge-following algorithms, like the Simplex
algorithm, start with a vertex of the feasible region and traverse along successive
edges of the region according to some prescribed rule, until an optimum vertex is


18
u
Figure 4. The two possibilities for a non-planar revisit of T to F where
F meets the boundary of the annulus in the case of the torus.
First consider the case in Figure 4a. above. In this case, t = u. Let t0 be as
shown. Replace T by the path Ti = F[if,s] U r[s,u] indicated by the dotted path
in Figure 4a. If Ti has only planar revisits, then we are done; so assume that Ti
has a non-planar revisit (si,ti) to a face Fi. Note that si and t\ cannot both lie on
Tifu, s] since this would mean that F and F\ meet improperly. Hence, without loss
of generality, assume that si lies on Fi(s, u] and t\ lies on Ti[u, s]. Among all choices
for Fi, choose the one for which Si is nearest to v along Ti. It is easy to see that it
suffices to consider the six possibilities for Fi shown in Figure 5.


58
Theorem 3.3.3. Every polyhedral map on N3 or S2 that has a non-separating
polygonal representation contains a non-planar, non-revisiting cycle.
Proof. Let M be a polyhedral map on N3 and assume that M has a non
separating polygonal representation P. Let the vertices of Tp be labeled 1, ...,w and
for i = l,...,u, let n be the number of occurrences of the label i on Tp. Since the
Euler characteristic in this case is 1, by Lemma 3.1.1,
n\ -f- ... 4- nv 2,v T 4. (3-3)
where n¡ > 3 for i = l,...,u. It is easily checked that the above equation has no
solutions for v > 5. That is to say, there are at most four different vertex labels on
Tp. Hence the graph Gp, obtained by identifying matched directed edges on Tp can
have at most four vertices and consequently, must be planar. By the same argument
given in the alternate proof to Theorem 3.2.1, the graph G obtained by identifying
matched edges on dP must be homeomorphic to Gp and consequently, must also be
planar. By Theorem 3.3.2, the conjecture is true for the graph G. Hence by Theorem
3.3.1, M has a non-planar, non-revisiting cycle that is contained in dP.
Next, consider the case where M is a polyhedral map on S2 and let P be a non
separating polygonal representation of M. Since the Euler characteristic in this case
is 2, by Lemma 3.1.1,
n\ -)-... T nv 2v -f- 6. (3-4)
where n,- > 3 for i = 1,..., v. It is easily checked that the above equation can have
solutions only if v < 6. If e is the number of distinct directed edges on Tp, then e < 9.
Let G be the graph obtained by identifying matched edges on Tp. Then G has at
most 6 vertices and at most 9 edges. Since K5 has 10 edges, G is either planar, or is
isomorphic to Ks. In either case, the coloring conjecture for two colors is true, and


2
graph of a polytope P is the one-dimensional skeleton of P. In particular, a theorem
of Steinitz and Rademacher [20] states that a graph is (isomorphic to) the graph of
a 3-polytope if and only if it is planar and 3-connected. By a generalization due to
Balinski [2], the graph of a d-polytope is d-connected. A directed graph consists of
a set of vertices and a set of ordered pairs of distinct vertices called directed edges.
A directed edge (u,v) is represented by an edge with endpoints u and v and an
arrowhead pointing towards v denoting the direction of the directed edge (u,v).
A surface S is a connected 2-dimensional manifold, possibly with boundary dS.
There are two kinds of closed surfaces, orientable and non-orient able. The 2-sphere,
torus, double torus, and so on are orientable while the projective plane, Klein bottle
and so on are non-orientable. It is well known that any orientable surface may be
obtained by attaching a suitable number of handles to the sphere, while any non-
orientable surface may be obtained by attaching a suitable number of Mobius bands
to the sphere. An orientable surface denoted by Sg is said have genus g, if one must
add g handles to the sphere to obtain its homeomorphism type. On the other hand,
a non-orientable surface denoted by Nk is said to have crosscap number k, if one must
attach k Mobius bands to the sphere to obtain its homeomorphism type. If Si and
S2 are surfaces without boundary, then their connected sum is the surface obtained
by removing the interior of a disk from Sj and S2 and then identifying the resulting
boundary components. Thus, the surface Sg is homeomorphic to the connected sum
of g tori, while the surface Nk is homeomorphic to the connected sum of k projective
planes. Let G be a connected graph embedded on a surface S such that GPdS = dS.
Then the pair (G, S) is called a map on S and is denoted by M. The vertices and edges
of M are those of G?, and the faces of M are the closures of the connected regions in
the complement of G on S. If G is embedded on the plane, then the map is called a
planar map. A planar map has exactly one unbounded face. If M = ((?, S) is a map


47
e Since / does not belong to A, f must belong to A. Also, since P is connected,
so is its dual P*. Also, by definition, P* fl dP 0. Hence, with v¡ and v j> as the
vertices of P* corresponding to the faces / and f of P, respectively, there is a path
Vf, u/j,..., V/k, Vj> from v¡ to ty in P* that is contained in the interior of the polygon
OP. Hence, there is a sequence of faces /, i = 1,..., k of P corresponding to the ver
tices Vft of P* such that fC\fi, f\ fl/2,..., fk-\ D /*, fk C\f are all edges of P that are
contained in the interior of the polygon dP. But the edges of C are all on dP. Hence
the interior edges /fl/x, /i fl/2,..., fk-1 H fk,fkPf', are all in the cell A. This implies
that the face / is also in A, which is a contradiction to the choice of / as a face of
M not in A. Hence C does not bound a cell on the surface and must be non-planar.
Corollary 3.2.1. Every polyhedral map on a surface (except on the sphere) has
a non-planar cycle.
Proof. Let M be a polyhedral map and let P be a polygonal representation of
M. By the definition of a polygonal representation, every vertex on dP appears at
least twice on dP. Hence there is at least one cycle that is contained in dP that
is obtained by traveling along dP between two consecutive vertices both labeled A
that have the property that there is no other pair of matched vertices that appear
between the A/s. By Lemma 3.2.1, such a cycle must be non-planar.
Theorem 3.2.1 Let M be a polyhedral map on the projective plane, torus, or
Klein bottle. If M has a non-separating polygonal representation, then M has a
non-planar, non-revisiting cycle.


13
that some of x,x',y', and y are the same vertex. Denote by Fn,..., Fm, m > n the
faces of M that meet H[x\y] but not x or y. With x" as the vertex of T not on H that
is incident to x, let Fx,Fn_i be the faces of M that meet H and lie in the region
determined by the sector with central angle x"xx Likewise, let Fm+1,F/v be the
faces that meet H and lie in the region determined by the sector with central angle
y'yy", where y" is the vertex of T not on H that is incident to y. For i = 1,N 1,
let Xi be the vertex on the edge F fl F,-+i that doesnt belong to H. Choose x0 on
r[it,ic) with the property that for some s 1,..., TV, the face Fs contains an edge
not in T that is incident to Xq. Such a vertex exists for otherwise one of the Fs has
a disconnected intersection with dS, which is a contradiction. Similarly, choose y0
on r(y,u] such that the face Ft,t > s, contains an edge not in T that is incident to
yo Note that x0 and y0 must lie in the interior of S for otherwise Fs or Ft has a
disconnected intersection with dS (meeting dS at both H and Xo or y0, respectively),
which is a contradiction. Construct a path from xo to yo that doesnt meet dS as
follows:
Since x0 and xs belong to Fs and lie in the interior of S, there must be a path
Fjxo,^] from x0 to xs along the face Fs that avoids dS, for otherwise Fs fl dS is
disconnected. For k = s,..., t 2, the vertices and lie on the face Fk+1 and
are in the interior of S. Hence by the argument above, for each k, there is a path
Fk+i[xk,xk+1] along Ffc+i that lies in the interior of S. And let Ft[xt-i,yo\ be the
interior path from xt-i to yo along Ft. By construction,
I =
t2
U Ffc+i[x,U Ftjxt-i, yo]
k=s
n 5F = 0.
In T, replace r[x0,yo] by /. Now there may be repeated vertices on T. In this case,
remove the vertices of T that appear between successive occurrences of each repeated
vertex, eventually yielding a path Tx from u to v where Tx C H = 0. If Tx lies in


27
Next consider the case where M is a polyhedral map on the Klein bottle and let
u and v be vertices of M. The proof in this case is similar to that in the case of
the torus with a few subtle differences. As before, use Lemma 2.1.4 to cut M along
a non-planar, non-revisiting cycle C yielding an annulus A with bounding cycles C\
and C2 and consider the following three cases:
1. u and v lie on C. In this case, the argument is identical to the one given above for
the torus.
2. u lies on C and v does not lie on C Without loss of generality, assume that u lies
on C\. As in the case of the torus, consider the path T joining u and v shown in
Figure 4. If T has only planar revisits, we are done, so assume that T has a non-planar
revisit (s,t) to a face F and among all non-planar revisits of F, choose the one for
which s is nearest to v along T. Consider the following cases :
i. F contains u It suffices to consider the cases shown in Figures 4a. and 4b. Replace
T by the path Ti = F[u,s] U r[s,u]. If Ti has only planar revisits, we are done; so
assume that Ti has a non-planar revisit to a face Fx. As in the case of the torus, such
a revisit must involve a vertex si of Ti(5, v] and a vertex of Tifii, 5]. Without loss of
generality, there are five possibilities for Fx as shown in Figure 11. In the cases in
Figures 11a, lib, and 11c, T has a non-planar revisit to Fi contradicting the choice
of F with s nearest to v. So consider the case in Figure lid and let t2 be as shown.
Replace Ti by the path T2 = Ti[w,2] U Fi[2,Si] U Fi[si,u]. It can be checked that
r2 can have only planar revisits. The case in Figure lie is similar to the analogous
case for the torus.
ii. F does not contain u. This case is similar to the analogous case for the torus.
3. Neither u nor v lies on C. Again, this case is similar to the analogous case for
the torus.
This concludes the proof of the theorem.


22


6
[18] has shown that the conjecture is false for a very general type of 2-dimensional
complex. Mani and Walkup [19] have shown that the conjecture is false for 3-spheres.
Barnette [5,7] has recently shown that the non-revisiting path conjecture is indeed
true for polyhedral maps on cell complexes that are homeomorphic to the projective
plane and the torus. Engelhardt [10] has shown in her Ph.D. dissertation that the
non-revisiting conjecture is also true for polyhedral maps on the Klein bottle. In a
recent paper [8], Barnette gives counterexamples to the non-revisiting path conjecture
that are polyhedral maps on the surfaces Ss and iV16.
Similar to the notion of a path in a polyhedral map having a disconnected inter
section with a face of the polyhedral map, one may consider a cycle in the underlying
graph of a polyhedral map that has a disconnected intersection with a face of the
polyhedral map. A cycle of a polyhedral map refers to a cycle in the underlying graph
of the polyhedral map. Let M (G, S) be a polyhedral map and C be a cycle in M.
Then C is said to be non-planar if it does not bound a cell on S. Suppose i2[s,i] is
a path along C from s to i such that for some face F, s and t are on F, and i?[s,t]
along with either path along F from s to t bounds a cell on S. Then the path F[s,t]
is called a planar revisit of C to the face F. A cycle is non-revisiting if it does not
have any revisits; in other words, for each face F of M, C fl F is either empty, or
connected. According to a theorem due to Barnette [6], if M has a non-planar cycle
all of whose revisits are planar, then M has a non-planar, non-revisiting cycle. It is
also known that every polyhedral map on a projective plane, torus, or Klein bottle
has a non-planar, non-revisiting cycle [6]. However, the problem of the existence of
such cycles on other surfaces is still open.
Using the result for the three surfaces mentioned above, Barnette [6] proves that
every polyhedral map on the torus is the union of two face-disjoint subcomplexes


29
u
e.
Figure 11. The five possibilities for a non-planar revisit of T to F
in the case in Figure 4a for the Klein bottle.
2.2 Counter-Examples to the Non-Revisiting Path Conjecture
Barnette has recently shown that there exist polyhedral maps on the surfaces
S& and Ah6 for which the non-revisiting path conjecture is false. In Engelhardts
dissertation, it is claimed that the non-revisiting path conjecture is also true for
polyhedral maps on the surface S2 In this section, we settle the non-revisiting path
conjecture for polyhedral maps on all surfaces except N3. Specifically, it is shown
that for each g > 2, the non-revisiting path conjecture is false for the surface Sg, and
for each k > 4, the non-revisiting path conjecture is false for the surface iV*. This


CHAPTER 1
INTRODUCTION
1.1 Basic Definitions
A polyhedron is the intersection of a finite collection of closed half-spaces in 77-
dimensional Euclidean space, and a polytope is a bounded polyhedron. Equivalently,
a polytope is the convex hull of a finite set of points in Euclidean space. If a polytope
is d-dimensional, then we say that it is a d-polytope. A face of a polytope P is
0, P itself, or the intersection of P with a supporting hyperplane. With prefixes
denoting dimension, the 0-, 1-, and (d l)-dimensional faces of a dpolytope are
called vertices, edges, and facets. A d-polytope P is simple if each of its vertices are
incident to precisely d edges, or equivalently, to d facets. A d-simplex is the convex
hull of d + 1 affinely independent points, and a polytope is called simplicial if each of
its facets is a simplex. There is a duality between the notions of simple and simplicial
polytopes. That is to say, there is a bijection between the set of simple d-polytopes
with n vertices and the set of simplicial d-polytopes with n facets that preserves
incidences and complements dimensions.
A graph G is a finite non-empty set of objects called vertices together with a
(possibly empty) set of unordered pairs of distinct vertices called edges. The vertex
set of G is denoted by V{G), while the edge set of G is denoted by E(G). A graph is
d-connected if the removal of fewer than d vertices yields neither a disconnected graph
nor the trivial graph. A cut-vertex of a graph is a vertex whose removal disconnects
the graph. Thus a graph is 2-connected if and only if it has no cut-vertices. The
1


Fij
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
LIST OF FIGURES
A simpler proof of a result due to Barnette 11
A non-planar revisit of T to F on the projective plane 15
A non-planar revisit of Ti to F\ on the projective plane 16
The two possibilities for a non-planar revisit of T to F where F meets
the boundary of the annulus in the case of the torus 18
The six possibilities for a non-planar revisit of Ti to Fi in the case in
Figure 4a 20-21
The three possibilities for a non-planar revisit of Ti to F\ in the case in
Figure 4b 23
A non-planar revisit of T to F where F does not meet the boundary of
the annulus 24
A non-planar revisit of Tx to Fy in the case in Figure 7 25
The two possibilities for a non-planar revisit of T2 to F2 in the case in
Figure 8 26-27
A non-planar revisit of T to F in the case where both vertices are in
the interior of the annulus 28
The five possibilities for a non-planar revisit of T to F in the case in
Figure 4a for the Klein bottle 30-31
The faces that constitute a counter-example to the non-revisiting path
conjecture for polyhedral maps on Si 33
An orientation on the faces in Figure 12 that shows that the surface is
orientable 34
The faces that constitute a counter-example to the non-revisiting path
conjecture for polyhedral maps on N4 36
An improper matching of edges on dP 40
An example of a polyhedral map M, a polygonal representation P of
M, and the type of M 41
Two polygonal representations of a polyhedral map on the torus.. .42
The types of polygonal representations of polyhedral maps on the
projective plane, torus, and Klein bottle 43-44
IV


REFERENCES
[1] Adler, I. (1974). Lower Bounds for Maximum Diameters of Polytopes, Pivoting
and Extensions. Math. Programming Study. 1, 11-19.
[2] Balinski, M. L. (1961). On the Graph Structure of Convex Polytopes in n-Space.
Pacific Journal Of Math. 11, 431-434.
[3] Barnette, D. W. (1969). Wv Paths on 3-Polytopes. J. Combinatorial Theory. 7,
62-70.
[4] Barnette, D. W. (1973). Graph Theorems for Manifolds. Israel Journal Of Math.
16, 62-72.
[5] Barnette, D. W. (1986). Wv Paths in the Projective Plane. Discrete Math. 62,
127-131.
[6] Barnette, D. W. (1988). Decomposition Theorems for the Torus, Projective
Plane and Klein Bottle. Discrete Math. 70, 1-16.
[7] Barnette, D. W. (1990). Wv Paths on the Torus. Discrete Comp. Geom. 5, 603-
608.
[8] Barnette, D. W. (1993). A 2-manifold of Genus 8 Without the LU-Property.
Geometriae Dedicata. 46, 211-214.
[9] Dantzig, G. B. (1963). Linear Programming and Extensions. Princeton Univer
sity Press, Princeton, N.J.
[10] Engelhardt, E. (1988). Some Problems on Paths in Graphs. Ph. D. thesis. Uni
versity of Washington, Seattle, Wa.
[11] Gross, J. L. and Tucker, T. W. (1987). Topological Graph Theory. Wiley Inter
science Series in Discrete Mathematics and Optimization.
[12] Kalai G. and Kleitman D. (1992). A Quasi-Polynomial Bound for the Diameter
of Graphs of Polyhedra. Bull. Amer. Math. Soc. 226(2), 315-316.
[13] Klee, V. (1965). Problem 19, Colloquium On Convexity. (Copenhagen).
[14] Klee, V. (1965). Paths on Polyhedra I. J. Soc. Indust. Appl. Math. 13, 946-956.
[15] Klee, V. (1966). Paths on Polyhedra II. Pacific Journal Of Math. 17, 249-262.
[16] Klee, V. and Walkup D. (1967). The d-Step Conjecture for Polyhedra of Dimen
sion, d < 6. Acta Math. 133, 53-78.
[17] Klee, V. and Kleindschmidt, P. (1987). The d-Step Conjecture and its Relatives.
Mathematics Of Operations Research. 12(4), 718-755.
[18] Larman, D. G. (1974). Paths on Polytopes. Proc. Bond. Math. Soc. 3, 161- 178.
61


NON-REVISITING PATHS AND CYCLES IN POLYHEDRAL MAPS
By
HARI PULAPAKA
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA


32
easy to check that the faces Fi,..., F16 meet properly. Hence Mi is a polyhedral map
on S2. It remains to be shown that M\ does not have the non-revisiting property.
We will show that the vertices labelled x and y cannot be joined by a non-revisiting
path in Mi. The proof is by contradiction; so assume that T is a non-revisiting path
joining x and y in M\. Without loss of generality, assume that the vertex incident to
x along T is the vertex labelled A (the proof is symmetric in the other cases). Note
that in this case, the path L has left the faces labelled Fs and F4. Furthermore, the
label A also appears on the face Fq. Since Y was assumed to be non-revisiting and
the vertex y lies on F6, the remainder of T must lie on the face F6. There are two
ways of getting from A to y along F6, namely, through the vertices labelled 2 or 3. If
T passes through the vertex 2, then the face F4 is revisited by T, which contradict
ing the assumption that T is a non-revisiting path. On the other hand, if T passes
through the vertex labelled 3, then T revisits the face F3; also a contradiction. Thus,
there can be no non-revisiting path from x to y in M.
In order to prove the result for the surface Sg,g > 3, we form the connected sum
of the surfaces S% and Sg~2 as follows:
Let Mi be a polyhedral map on the surface Sg-2 such that Mi has a triangular face
T. Assign the same labelling on the vertices of T as the face F9 of Mi. Glue the
polyhedral maps Mi and Mi by identifying the faces F9 and T. Then remove this
face from the cell complex. The result is a map Mi on the surface Sg. In fact, Mi"
is a polyhedral map on Sg. In order to prove this, it suffices to show that the faces of
Mi meet properly. Let F and G be faces of Mi. If F and G are also faces of Mi,
or Mi then they clearly meet properly. Without loss of generality, assume that F is
a face of Mi and G is a face of Mi The only way that they can meet in Mi is if F
meets F9 in Mi and G meets T in Mi In this case, F and G have to meet properly


53
Theorem 3.3.1. If the conjecture is true in the case where each edge is colored
with exactly two colors, then every polyhedral map with a non-separating polygonal
representation contains a non-planar, non-revisiting cycle.
Proof. Let P be a non-separating polygonal representation of a polyhedral map
M on a surface. Let {F}, i = 1 ,...,&, be the collection of faces of M that have at
least one edge in common with dP. Since P has no separating faces, for i 1,k,
Pi = Fi fl dP is a path in dP. For i color the edges of the path Pi using
a distinct color C\. Since the edges on dP are matched in pairs, every edge in M
that lies on dP is colored using exactly two colors. Now consider dP and identify
the matched edges on dP. The result is a graph G, where each edge is colored using
exactly two colors. Note that the cycles in G are the cycles of M contained in dP.
Recall, by Lemma 3.2.1, that the cycles that are contained in dP are non-planar.
Furthermore, there is a monochromatic cycle in G if and only if, for some i, two
vertices of Pi are identified. This in turn implies the face Fi of M is not simply-
connected which is impossible. There is a dichromatic cycle in G if and only if there
are faces Ft and Fj of M that meet improperly which is also not allowed. Finally, a
cycle in G has a monochromatic chord using a color (7 if and only if the corresponding
cycle on dP revisits the face Fi of M. Hence, if the conjecture is true, then there
must be a cycle of M contained in dP that has no monochromatic chord and hence
must be non-revisiting.
Example 3.3.1. The conjecture is true for all graphs that contain a triangle.
Proof. Let G be a graph and T a triangle of G. The proof is by contradiction;
so assume that G has an edge-coloring with no mono or dichromatic cycles such that
every cycle of G has a monochromatic chord. In particular, T has a monochromatic
chord P. Since there are no multiple edges between vertices, P has length at least


57
inserting vertices of G that are not in Gt along the interior of each edge of Gj. Thus
Gt and G are homeomorphic. If M is a polyhedral map on the projective plane, then
by Theorem 3.1.2, P is of type I and consequently, Gt is isomorphic to the graph in
Figure 28a. By a similar argument, if M is a polyhedral map on the torus, then G is
homeomorphic to the graph in Figure 28b, or Figure 28c. and if M is a polyhedral
map on the Klein bottle, then G is homeomorphic to the graph in Figure 28b, Figure
28c, or Figure 28d.
CShiO
d.
Figure 28. The boundary graphs of polyhedral maps on the
projective plane, torus, and Klein bottle.
In all cases, G is planar and by Theorem 3.3.2, the conjecture is true. Hence by
Theorem 3.3.1, in each case, there is a non-planar, non-revisiting cycle in M.
So far, two elementary proofs of Theorem 3.2.1 have been provided. However, the
scope of Theorems 3.3.1 and 3.3.2 are greater than merely giving proofs for already
known results on non-planar, non-revisiting cycles on the projective plane, torus,
and Klein bottle. In this regard, the following result extends Barnettes result on
non-revisiting cycles on the three surfaces mentioned above to a class of polyhedral
maps on the surfaces IV3 or Si.


54
two. Let Vi and v2 be such that P fl T {ui,n2} and e = v\v2 the edge in T. Then
the cycle P[vi, v2\ U {e} is a mono or dichromatic cycle, which is a contradiction.
Example 3.3.2. The conjecture is true for K3j3.
Proof. The proof is by contradiction; so assume that there is an edge-coloring of
K3j3 with no mono or dichromatic cycles such that every cycle has a monochromatic
chord. Let the vertices of K3}3 be labeled as shown in Figure 27.
1
Figure 27. The graph-coloring conjecture is true for K3t3.
It will be shown that there must be a 4cycle with no monochromatic chord. It
is easy to see that a monochromatic chord of any 4cycle must have length at least
two. If a monochromatic chord of a 4cycle has length greater than two, then the
endpoints of the chord are adjacent to each other in K3t3. This is a contradiction since
in this case, the monochromatic chord together with the edge joining the endpoints
of the chord form a mono or dichromatic cycle. Hence a monochromatic chord of
a 4cycle must have length exactly two. First, consider the 4-cycle Cj = 1 A3C1
and let Pi be a monochromatic chord of C\. Up to symmetry, Pi = C2A. Next,
the cycle C2 IA2CI must have a monochromatic chord P2. The vertices labeled
A and C cannot be the endpoints of P2 for then, the paths Pi and P2 form a mono
or dichromatic cycle. Hence P2 = 1P2. Let C3 = IA2BI and P3 a monochromatic
chord of C3. By a similar argument as the one given for P2, P3 = A3B. If P4


12
If x = v, we are done, otherwise we will obtain a contradiction. Let (z,y) be a
revisit of the path Fo[ii,x] U r[x,u] to a face F of M. By statement (3), z r0[u,x]
and y 6 r[x,u]. Among all revisits by this path we choose F so that 2 is nearest to
u along ro[w,x]. Now consider the path I\ = ro[u,z] U F[z,y\ U r[y,u] from u to v
(indicated by the dotted path in Figure 1) and observe the following :
(i) Ti is a path from u to v all of whose revisits are planar. To see this note that
To itself has no revisits. A revisit involving vertices of F[y, w] alone has to be planar
since T has only planar revisits. A non-planar revisit by Ti cannot involve vertices
of F[z,y] since the closed path F[z,y\ U r[y, x] U r0[x,z] bounds a cell. Finally, if a
revisit by Ti involves a vertex of L and a vertex of To, then it must be planar since
r0[u, x] U r[x, u] admits only planar revisits.
(ii) ri[u,y] does not revisit any face of M. A revisit by Fi[u,y] to a face Fi must
involve y and a vertex z of Fo[u,z). Note that z ^ z; otherwise F and Fi meet
improperly at y and z. Now (z,y) is a revisit of the path r0[u,x] U r[x,u]. This
contradicts the choice of F with z nearest to u on To[u,x].
The existence of y contradicts the choice of x as the vertex that was furthest along
r[u,u] satisfying conditions (1) and (2).
Lemma 2.1.2. Let S be a surface with boundary dS and M = (G, S) a polyhedral
map on S such that the intersection of any face of M with dS is either empty, or
connected. Then any two vertices of M that lie in the interior of S can be joined by
path in M that is contained in the interior of S.
Proof : Since G is connected, there is a path T from u to v in M. If T lies in
the interior of S, we are done; so assume that T fl dS 7^ 0 and let H = r[x,y] be a
connected component of T fl dS with the order of vertices along T being u,x,y,v. Let
x and y be the vertices of H that are incident to x and y, respectively. It is possible


24
is for the non-planar revisit to involve a vertex si of Ti(s, t) and u as shown in Figure
8.
u
Figure 8. A non-planar revisit of Ti to Fx in the case in Figure 7.
Among all choices for F%, choose the one for which Si is nearest to v along I\
and replace Ti by the path T2 = si] U Fi[s!,u] as shown in Figure 8. Now there
are two possibilities for a non-planar revisit of T2 to a face These are shown in
Figures 9a and 9b. In both cases, among all such choices for F2, choose the one for


16
Again, if Ti has only planar revisits, we are done. On the other hand, if rx has
a non-planar revisit to a face Fi, then it must involve a vertex sx of rx(s,u] and a
vertex of rx[u,f] as shown in Figure 3.
t *2
Figure 3. A non-planar revisit of Tx to Fi on the projective plane.
Among all choices for Fi, choose the one for which sx is nearest to v along rx and
let t2 be as shown. Replace rx by the path T2 = rx[ti,t2] U Fx[2,,sx] U rx[sx,i;]. It
can be easily checked that T2 can have only planar revisits.
3. Neither u nor v lies on C. In this case, both u and v lie in the interior of the cell
if. Hence by Lemma 2.1.2, there is a path T joining u and v that is contained in the


TABLE OF CONTENTS
ACKNOWLEDGEMENTS ii
LIST OF FIGURES iv
NOTATION vi
ABSTRACT vi
CHAPTERS
1 INTRODUCTION 1
1.1 Basic Definitions 1
1.2 Some History and Motivation 3
1.3 Non-Revisiting Paths and Cycles 5
1.4 A Summary of the Research 7
2 NON-REVISITING PATHS ON SURFACES 10
2.1 Non-Revisiting Paths on the Projective Plane, Torus, and Klein Bottle 11
2.2 Counter-Examples to the Non-Revisiting Path Conjecture 29
3 NON-REVISITING CYCLES ON SURFACES 35
3.1 Polygonal Representation of Polyhedral Maps 35
3.2 Polygonal Representation and Non-Revisiting Cycles 46
3.3 A Graph-Coloring Problem and Non-Revisiting Cycles 52
4 CONCLUSION 60
REFERENCES 61
BIOGRAPHICAL SKETCH 63
m


8
the non-revisiting cycle. The proof also utilizes an important lemma due to Barnette
which states that a path with only planar revisits can be modified to a non-revisiting
path. This result plays a key role even in the earlier proofs. In Section 2.1, a simpler
proof of this lemma is given. Section 2.2 deals with the non-revisiting path conjecture
for the other surfaces. In Engelhardts dissertation, a proof of the validity of the non
revisiting path conjecture for the surface S2 is given. In Section 2.2, it is shown that
this is impossible! In fact, it is shown that the non-revisiting path conjecture is false
for polyhedral maps on the surfaces Sg,g > 2, and Nk, k > 4. Thus, the non-revisiting
path conjecture for polyhedral maps is settled for all surfaces except IV3.
The second problem concerns the existence of a non-planar, non-revisiting cycle
in a polyhedral map. As stated earlier, the only surfaces that are known to contain
such cycles are the projective plane, torus, and Klein bottle [6]. In this context, in
Section 3.1, the notion of a polygonal representation of a polyhedral map is defined.
This is analogous to that of representing a surface as a polygon whose sides are
directed and matched in pairs. Except, in the case of a polygonal representation of
a polyhedral map, the representation preserves the combinatorial structure of the
underlying graph of the polyhedral map. In other words, a polygonal representation
is a polyhedral map on a closed disc with certain matching conditions on the edges
of the polyhedral map that lie on the boundary of the disc. It is shown that every
polyhedral map has a polygonal representation. Next, the notion of a non-separating
polygonal representation of a polyhedral map is defined. As will be evident from its
definition, the existence of such a representation is a rather desirable property of
a polyhedral map. An interesting question is: Which polyhedral maps have a non
separating polygonal representation ? In this regard, it is shown that there exist an
infinite family of polyhedral maps that do not possess a non-separating polygonal
representation. Elementary Euler Characteristic arguments allow the enumeration of


19
u u
c.
d.


Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
NON-REVISITING PATHS AND CYCLES IN POLYHEDRAL MAPS
By
Hari Pulapaka
August 1995
Chairman: Dr. Andrew Vince
Major Department: Mathematics
In this dissertation, three problems are considered. The first problem is related to
one of the most famous unsolved problems in the combinatorial theory of polytopes
called the Hirsch conjecture, which proposes a bound on the diameter of the graph of
a polytope. An equivalent conjecture due to Klee and Wolfe, called the non-revisiting
path conjecture, asserts that any two vertices of a polytope can be joined by a path
that does not revisit a facet. The non-revisiting path conjecture can be extended to
cell complexes that are more general than those that are the boundary complexes
of polytopes. In this regard, the non-revisiting path conjecture is known to be true
for polyhedral maps on the 2sphere, projective plane, torus, and Klein bottle. In
this research, an elementary, unified proof of the validity of the non-revisiting path
conjecture for polyhedral maps on the projective plane, torus, and Klein bottle is
given. In addition, it is shown that for polyhedral maps, the non-revisiting path
conjecture is false for all other surfaces except possibly surfaces homeomorphic to the
connected sum of three projective planes.
vn


19 The two possibilities for a 2connected, planar map with 14 vertices
and 3 hexagonal faces 48
20 Polygonal representations for the polyhedral maps in Figures 12 and
14 49
21 Non-planar, non-revisiting cycles on the projective plane 52
22 Non-planar, non-revisiting cycles on the torus 53
23 Non-planar, non-revisiting cycles on the torus 54
24 Non-planar, non-revisiting cycles on the Klein bottle 54
25 Non-planar, non-revisiting cycles on the Klein bottle 55
26 Non-planar, non-revisiting cycles on the Klein bottle 56
27 The graph-coloring conjecture is true for /f3i3 59
28 The boundary graphs of polyhedral maps on the projective plane, torus,
and Klein bottle 62
v


38
Figure 16
Figure 16. An example of a polyhedral map M, a polygonal representation P of M,
and the type of M.
a bc defga a
a.
b.
Figure 17. Two polygonal representations of a polyhedral map on the torus.


7
that are annuli. Similar decomposition theorems are proved for the projective plane
and Klein bottle.
1.4 A Summary of the Research
Primarily, three problems are considered in this dissertation. The first problem
is related to the non-revisiting path conjecture due to Klee and Wolfe. In its original
formulation, the non-revisiting path conjecture was in the context of convex polytopes.
A generalization due to Klee [13] of the conjecture led to the problem of the existence
of non-revisiting paths between any two vertices of a polyhedral map on a surface.
Chapter 2 deals with this question. Specifically, in Section 2.1, an elementary, unified
proof of the non-revisiting path conjecture for polyhedral maps on the projective
plane, torus, and Klein bottle is given. Although these results are already known,
the earlier proofs for the three surfaces are quite different. The proof given here uses
a result due to Barnette, which states that any polyhedral map on the projective
plane, torus and Klein bottle has a non-planar, non-revisiting cycle. Furthermore, in
the case of the torus and Klein bottle, cutting along this non-revisiting cycle yields an
annulus. Hence in the case of the projective plane, the polyhedral map is cut along a
non-revisiting cycle yielding a cell whose boundary corresponds to the non-revisiting
cycle and the arguments presented pertain to the cell thus obtained. In the case of the
torus and Klein bottle, the arguments pertain to an annulus whose bounding cycles
correspond to the the non-revisiting cycle in the polyhedral map. The unification
of the proofs for the three surfaces is obtained by considering the same basic cases
for all three surfaces, namely either both vertices lie on the non-revisiting cycle, one
lies on the non-revisiting cycle and one does not, or neither of the two vertices lie on


50
A
Figure 23. Non-planar, non-revisiting cycles on the torus.
A t
B
Figure 24. Non-planar, non-revisiting cycles on the Klein bottle.
In both cases, replace C by the cycle C\ = (B, B) as shown. Now C\ must be
non-planar and non-revisiting.
If P is of type V, then by the same argument as for the projective plane the cycle
C shown in Figure 25 is non-planar and non-revisiting. If P is of type VI or type
VII, then the cycles shown in Figures 26a and 26b, respectively can be easily checked
to be non-planar and non-revisiting.


4
reached. The d-step conjecture and its relatives (including the Hirsch conjecture)
play a crucial role in the study of the computational complexity of such edge-following
algorithms. The d-step conjecture, formulated by W. M. Hirsch in 1957 and reported
in 1963 by Dantzig in his book Linear Programming And Extensions [9], has several
equivalent forms. One version, dealing with the maximum diameter A(d, n) of (the
graphs of) d-dimensional polytopes with n facets, asserts that A(d, 2d) = d for all
d while the Hirsch conjecture asserts that A(d, n) < n d for all n > d > 2. It
was proved by Klee and Walkup [16] that the d-step and the Hirsch conjectures are
equivalent, though not necessarily on a dimension to dimension basis. The distance,
dp(u, v) between two vertices u and v of a polytope P is a lower bound on the
complexity of applying an edge-following algorithm to P with initial vertex u and
target vertex v. Thus A(d, n) is a lower bound for the worst-case behaviour of edge
following LP algorithms over all d-polytopes with n facets. Since this applies to
all edge-following algorithms, A(d, n) estimates the worst possible behaviour of the
best possible edge-following algorithm. The d-step and Hirsch conjectures remain
unsettled, though they have been proved in many special cases, and counterexamples
have been found for slightly stronger conjectures. Specifically, the d-step conjecture
has been proved for d < 5. Although sharper results are known for small values of d
and n d, the best known general bounds for A(d, n) are due to Adler[1] and Kalai
and Kleitman [12], respectively. They are as follows :
Ln d + 1 < AK n) < nlosd+2.
It is generally believed that the d-step and Hirsch conjectures are false. However,
finding counterexamples to that effect would be merely a small first step in the line of
investigation related to the two conjectures. For the recent status of the conjectures


40
B
B
B
Figure 18. The types of polygonal representations of polyhedral maps
on the projective plane, torus, and Klein bottle.


5
and their relatives, the survey paper by Klee and Kleindschmidt [17] provides an
excellent source.
1.3 Non-Revisiting Paths and Cycles
If T is a path in a polyhedral map M, a revisit of F to a face F is a pair of vertices
(x,y) such that T[x,j/] flF = {x,y} where r[x,y] is the path along F from x to y.
Let (:r, y) be a revisit of a path T to a face F. If the two paths along F from x to y
are denoted as F[x,y] and F[x,y], then the revisit (x,y) is said to be planar if either
F[x,y\ U T[x,y] or F[x,y] U Y[x,y] bounds a cell on the surface. (Note that if one
does then so does the other.). A path is non-revisiting if it has no revisits.
In research on the d-step and Hirsch conjectures, it has been found that the
conjectures can be stated in several equivalent forms (even though no solution to
any one of them seems to be in sight!). One equivalent formulation is in terms of
the existence of non-revisiting paths in the graphs of convex polytopes. Part of this
research is related to the following non-revisiting path conjecture of Klee and Wolfe
(also called the Wv conjecture): Any two vertices of a polytope P can be joined by
a path that does not revisit any facet of P. Despite an apparent greater strength of
this conjecture (which prompted its original formulation), it is known [15] that the
non-revisiting path conjecture is equivalent to the Hirsch conjecture.
If P is a 3-polytope, then the faces of P form a polyhedral map on the 2-sphere
and the validity of the non-revisiting path conjecture along with some strengthened
forms of the non-revisiting path conjecture are proved [3,14,15]. Klee [13] conjectured
that the non-revisiting path conjecture might be true for cell complexes that are more
general than the boundary complexes of convex polytopes. In this regard, Larman


11
2.1 Non-Revisiting Paths on the Projective Plane, Torus, and Klein Bottle
Although the proof of the validity of the non-revisiting path conjecture for poly
hedral maps on the projective plane, torus, and Klein bottle given here has many
details, the ideas involved are quite elementary.
First, a simpler proof of an important lemma originally due to Barnette [5] is
presented.
Lemma 2.1.1 Let M be a polyhedral map with vertices u and v. If there is a path
in M joining u and v all of whose revisits are planar, then there is a non-revisiting
path between u and v.
Proof. Let T[u, u] be a path in M all of whose revisits are planar. If T[u, v] is not
a non-revisiting path, then there is a vertex x on T[u, u] with the following properties:
(1) There is a non-revisiting path ro[u,x] between u and x.
(2) The path r0[u,x] U rfaqw] has only planar revisits.
A path satisfying (1) and (2) exists; simply take x = u.
(3) Among all choices for x satisfying (1) and (2), choose the one which is furthest
along the path r[u,u].
Figure 1. A simpler proof of a result due to Barnette.


21
In the cases in Figures 5a, 5b, and 5c, T has a non-planar revisit to the face Fi
contradicting the choice of F with s nearest to v along T. Consider the case in Figure
5d and let t\ be as shown. Replace Ti by the path r2 = Tiju, x] U Fi[H, Si] U Tifsi, u]
as shown in Figure 5d. Now T2 has only planar revisits, Next consider the case in
Figure 5e and replace Ti by the path T2 = Fifi,^] U rx[si,u], where F\[it,si] is a
path from u to si along Fi that meets dA at only u. Again, it can be checked that
r2 can have only planar revisits. And in the case in Figure 5f, replace Ti by the
path r2 = ri[u, in the interior of A. Now the only possibility for a non-planar revisit of T2 to a face
F2 is for it to involve a vertex s2 of r2[ii,Si] and a vertex t2 of r2[u,i0]- Among
all such choices for F2, choose the one for which s2 is nearest v along T2. Now
r3 = r2[u,2] U F2[t2,s2\ U r2[s2,u] can have only planar revisits.
Next consider the case in Figure 4b. Replace T by the path Ti F[u, s] U r[s,u]
indicated by the dotted path in Figure 4b. Again, if Ti has only planar revisits, we are
done. So assume that Ti has a non-planar revisit (si,ti) to a face F\. Topologically,
there are three possibilities for the face Fi. Without loss of generality, assume that
the three possibilities for Fi are as shown in Figure 6.
In the cases in Figures 6a. and 6b, T has a non-planar revisit to Fi which con
tradicts the choice of F with s nearest to v along T. Hence Fi must be as shown in
Figure 6c. Among all choices for Fi, choose the one for which si is nearest to v along
Ti. Replace T* by the path T2 = Tx[w, H] U Fi[i, si] U Tifsi, v\ where Fi[tl5 sj] is a
path along Fi from sj to t¡ that avoids Ci and C2 except possibly meeting C\ at u
in the case where tx = u. Such a path exists for otherwise, C revisits Fi which is a
contradiction. It can now be checked that T2 can have only planar revisits.


56
(1) Ck+1 U Bk+i Ck-
(2) Ck+1 H Bk+1 = Pk+1-
Conditions (1) and (2) above are equivalent to the statement that P+1 is contained
in Ck- The proof of the above claim is by induction on k. It was shown above that
the claim is true for k = 1. Assume that the claim is true for each i < k. If Pkpi, a
monochromatic chord for dCk, is contained in Ck, we are done, so assume that Pk+i
leaves Ck- However, if Pk+i does not leave Ck-1, then the only way it can return to
Ck is by crossing the monochromatic chord Pk twice giving a dichromatic cycle in G,
which is not allowed. On the other hand, if Pk+i leaves both Ck and Ck-1, then it
eventually has to return to Ck- But by condition (1) for the component Ck-i, this
means that it also returns to Ck-1- Thus, there is a subpath Qk+i of Pk+i that is
a monochromatic chord for the cycle dCk-i and that returns to Ck-1 after leaving.
This is a contradiction to the induction hypothesis. Hence Pk+1 cannot leave Ck,
proving the claim.
It follows that G would have to be an infinite graph in order that every cycle in G
have a monochromatic chord, which is a contradiction. Hence there can be no planar
graph that can be a counter-example to the conjecture and the theorem is true for
all planar graphs.
As an application of Theorem 3.3.1, we give another simple proof of Theorem
3.2.1
An alternate proof of Theorem 3.2.1 In the case of each surface, let P be
a non-separating polygonal representation of M and let Gt and G be the graphs
obtained by identifying the edges on Tp and dP, respectively. Since Tp is obtained
from dP by performing concatenations along dP, G can be obtained from Gp by


28
c.
d.


25
which s2 (as shown in Figures 9a and 9b) is nearest to v along T2 and let t2 be as
shown. Next replace T2 by the path T3 T2[u,t2] U F2[t2,s2] U r2[s2,u]. It can be
checked that T3 can have only planar revisits.
The case where F does not meet C2 only is similar to the case above. If F neither
meets C\ nor C2, the proof is again similar to the one given above.
3. Neither u nor v lies on C In this case, u and v lie in the interior of the annulus A.
By Lemma 2.1.2, u and v can be joined by a path T that is contained in the interior
of A. The proof that T can be modified to a path joining u and v that has only
planar revisits is identical to the one for Figure 7. Thus in all cases, u and v can be
joined by a path in M that has only planar revisits and by Lemma 2.1.1, we are done.
u
a.


41
Theorem 3.1.2. Let M (G,S) be a polyhedral map.
(1) If S is a projective plane, then M has a polygonal representation of type I in
Figure 18.
(2) If S is a torus, then M has a polygonal representation of type II or type III
in Figure 18.
(3) If S is a Klein bottle, then M has a polygonal representation of type IV, type
V, type VI, or type VII in Figure 18.
Proof. Consider the map M' with one face (the polygon P itself) obtained by
identifying matched directed edges on dP. Let v be the number of vertices and e the
number of edges on M'. Denote the vertex labels on dP by 1,2,..., v. Further, let n
denote the number of occurrences of the label i on dP. First consider the case where
M is a polyhedral map on the projective plane. By Lemma 3.1.1, v 1, n,- = 2; and
P is of type I. Next consider the case where M is a polyhedral map on the torus or
Klein bottle. Since x = 0 in this case, by Lemma 3.1.1, > 3 for i = 1, ...,v and
n\ ... -(- nv 2v T 2. (3-2)
Since equation (3.2) has no solutions for v > 2, v 1, or v = 2. Consider the
following cases:
(1) v = 1 : In this case, there is exactly one vertex label on Tp and P must be of
type III in the case of the torus and of type V or type VI in the case of the Klein
bottle.
(2) v = 2 : In this case two vertex labels A and B appear exactly three times on Tp.
Furthermore, P must be of type II in the case of the torus and of type IV or type
VII in the case of the Klein bottle.


23
(ii) F does not contain u. Recall that F can meet at most one of C\ or C2 First
consider the case where F does not meet C\. Since F does not contain u, there must
be path along F from s to i that is contained in the interior of A. Without loss of
generality, assume that F is as shown Figure 7.
u
Figure 7. A non-planar revisit of T to a F where F does not meet
the boundary of the annulus.
In this case, replace T by the path Tx = T[u, t] U F[t, s] U r[s, v] where F[t, 5] is a
path along F from s to t that is contained in the interior of A. If Ti has only planar
revisits, we are done; so assume that Ti has a non-planar revisit to a face F\. If this
non-planar revisit involves a vertex of ri(s, u] and a vertex of Tifn, t], then the proof
is identical to the one given for Figure 6c. It can be checked that the only possibility



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39
Lemma 3.1.1. Let M be a polyhedral map on a surface of Euler characteristic
X / 2, Tp the type of any polygonal representation of M, andv the number of distinct
vertex labels on Tp. If the vertices on Tp are labeled l,...,u and ni is the number of
occurrences of the label i on Tp, then rai + ... + = 2v + 2 2x- Furthermore, if
X 7^ 1, then ni > 3 for i = 1,..., v.
Proof. First note that there cannot exist a vertex label that appears exactly
twice on Tp except in the case where M is a polyhedral map on the projective plane.
To see this, suppose B is a vertex label that appears exactly twice on Tp and let
A be another vertex label such that (A, B) is a directed edge on Tp and (A, B)
its matching edge on Tp. If B = A, then there are no more vertex labels on Tp.
Hence the directed edge can be matched in exactly one way on Tp, and in this case
the surface is a projective plane. Next assume B ^ A. If there are no more vertex
labels on Tp, then either the directed edges (A, B) and (£?, A) can be concatenated
contradicting the fact that Tp is the type of a polygonal representation of M, or
the directed edge (B,B) cannot be matched on Tp, which is again a contradiction.
Hence, there must be another vertex with label C (possibly A) such that (B, C) is a
directed edge on Tp. Since the vertex label B appears exactly twice on Tp, there is
only one possibility for the matching edge (B,C) But the directed edges (A, B) and
(B,C) can be concatenated, which is a contradiction. Hence n > 3 for i = l,...,u.
Next, consider the map M' with one face (the polygon Tp itself) obtained by
identifying matched directed edges on Tp and let e be the number of edges in M'.
Since the directed edges are matched in pairs on Tp, e = 2ii^rh¡. fo follows from
the Euler formula v e + / = x that
ni + + nv 2u + 2 2x
(3.1)


9
all the types of polygonal representations of polyhedral maps on the projective plane,
torus, and Klein bottle. In Section 3.2, an elementary, unified proof of Barnettes
result [6] on non-revisiting cycles on the projective plane, torus, and Klein bottle for
polyhedral maps that possess a non-separating polygonal representation is given.
Motivated by the second problem, a graph-coloring conjecture is proposed in
Section 3.3. This is the third problem considered. It is shown that if the graph
coloring conjecture is true in a special case, then every polyhedral map that has a
non-separating polygonal representation, in fact, has a non-planar, non-revisiting cy
cle. It is shown that the graph-coloring conjecture is true for all graphs that contain
a triangle, all planar graphs, and AT,3- As a consequence, it follows that every poly
hedral map on a surface homeomorphic to AT, k = 1,2,3, and Sg,g = 1,2 that has a
non-separating polygonal representation, contains a non-planar, non-revisiting cycle.
This extends Barnettes result [6] to this class of polyhedral maps on the surfaces N3
and S2.


I certify that I have read this study and that in my opinion it conforms to accept
able standards of scholarly presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.
Irew Vince, Chairman
Vofessor of Mathematics
I certify that I have read this study and that in my opinion it conforms to accept
able standards of scholarly presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.
Krishnaswami Alladi
Professor of Mathematics
I certify that I have read this study and that in my opinion it conforms to accept
able standards of scholarly presentation and is fMlf adequ^t&^in scope and quality,
as a dissertation for the degree of Doctor of PhHosophj
Tirnothy DaVis
AssistanVProfessor
Information Sciences
Computer and
I certify that I have read this study and that in my opinion it conforms to accept
able standards of scholarly presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.
mosopny.
JTlusr'
Bernard Mair
Associate Professor of Mathematics
I certify that I have read this study and that in my opinion it conforms to accept
able standards of scholarly presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philos^
Neil L/white
Professor of Mathematics
This dissertation was submitted to the Graduate Faculty of the Department of
Mathematics in the College of Liberal Arts and Sciences and to the Graduate School
and was accepted as partial fulfillment of the requirements for the degree of Doctor
of Philosophy.
August 1995
Dean, Graduate School


43
7 = 0 and the surface is the 2sphere. If n 5 or 6, then Kn embeds on the torus.
However, the embedding is not a triangulation of the torus. So assume that n > 7.
If n = 0,3,4, 7(mod 12), then (n~3)(n~4) js an integer and any embedding of Kn on
S-f is, in fact, a triangulation of 57. To see this, let v, e, and / be the number of
vertices, edges, and faces respectively, of Mn. Then v = n and e Hence
by the Euler equation for 57, f = |e. Hence the embedding is a triangulation of 57.
Since there are no multiple edges between vertices, the faces of the embedding meet
properly and the map Mn (Kn, S'7) is a polyhedral map.
Claim. For each n as above, if Mn has a non-separating polygonal representation
Pn, then Pn* is contained in Mn* and has the following properties:
(1) The faces of Pn* are (n 1)gons.
(2) Pn* has either two, three, or four faces.
(3) The graph of Pn* is a planar, spanning, 2connected subgraph of the graph of
M*.
Proof of Claim. Statement (1) is obvious. The vertices of Pn that lie in the interior
of the polygon dPn span a complete subgraph of Kn that is also contained in the
interior of the polygon dPn. If the number of vertices of Pn that lie in the interior of
the polygon dPn is greater than four, then by the previous statement, the graph of Pn
would be non-planar, which is a contradiction since Pn is a planar map. Consequently,
P* can have at most four faces. If Pn* has no faces, then Pn must be separating,
which is a contradiction. If Pn* has exactly one face, then Pn must have faces that
meet improperly, which is also a contradiction since Pn is a polyhedral map. To see
statement (3), note that the graph of Pn* is planar, has all the vertices of M*, and
is 2connected because Pn was assumed to be non-separating.
Hence, for each n > 7, if Pn* has exactly two faces, then the graph of Pn* has
2n 6 vertices. On the other hand, if Pn* has exactly three faces, then the graph of


49
Figure 22. Non-planar, non-revisiting cycles on the torus.
If C has a revisit to a face F, then it can be easily checked that F must contain
both A and B for otherwise dP would have to revisit F which contradicts the as
sumption that P is non-separating. Up to symmetry, F must be as shown in Figure
22. Replace C by the cycle C\ = (B,B) as shown. If Ci revisits a face Fi, then it
can be checked that F\ must also contain the vertices labelled A and B. But this
means that F and Fj meet improperly at A and B which is a contradiction. Hence
Ci must be non-revisiting. Next, consider the case where P is of type III and let C
be the cycle (A, A) along dP as shown in Figure 23. By the same argument as in the
case of the projective plane, it is easy to see that C is non-planar and non-revisiting.
Finally, consider the case where M is a polyhedral map on the Klein bottle and
let P be a polygonal representation of M that is of type IV. Let C (A, A) be
the cycle shown in Figure 24. If C is non-revisiting, we are done, so assume that C
revisits a face F. As in the case of the torus, it can be checked that F must contain
both A and B. There are two possibilities for F as shown in Figure 24.


14
the interior of S, we are done; otherwise perform the same modification as above to
a connected component of rfi fl dS. When it is no longer possible to perform any
modifications, the result must be a path from u to v that is contained in the interior
of S. E
The following two lemmas are due to Barnette [6].
Lemma 2.1.3. Every polyhedral map on the projective plane has a non-planar,
non-revisiting cycle.
Lemma 2.1.4. Every polyhedral map M on the torus or Klein bottle contains a
non-revisiting cycle C such that cutting M along C yields an annulus.
We are now in a position to state and prove the main result of this section.
Theorem 2.1.1. Any two vertices of a polyhedral map M on the projective plane,
torus or Klein bottle can be joined by a non-revisiting path.
Proof : For each surface, we will show that any two vertices u and v can be
joined by a path in M all of whose revisits are planar. Consequently, by Lemma
2.1.1, there is a non-revisiting path joining u and v.
First consider the case where M is a polyhedral map on the projective plane. By
Lemma 2.1.3, M has a non-revisiting cycle C such that cutting M along C yields
a cell H whose boundary corresponds to the cycle C. Without loss of generality,
consider the following cases:
1. u and v lie on C. In this case, either of the two paths along C from u to v must
be non-revisiting (since C is non-revisiting).


CHAPTER 3
NON-REVISITING CYCLES ON SURFACES
This chapter consists of three sections. In Section 3.1, the notion of a polygonal
representation of a polyhedral map is introduced. As will be seen, this is a convenient
way to represent a polyhedral map as a polygon in the plane. It is shown that every
polyhedral map on a surface has such a representation and some useful properties
of polygonal representations are proved. The notion of a non-separating polygonal
representation is defined. An interesting question is: Which polyhedral maps have a
non-separating polygonal representation ? It is shown that not all polyhedral maps
have a non-separating polygonal representation. In Section 3.2, polygonal represen
tations are used to provide a simple, unified proof of the existence of a non-planar,
non-revisiting cycle in a polyhedral map on the projective plane, torus, and Klein
bottle. This is done for polyhedral maps that have a non-separating polygonal rep
resentation. And in Section 3.3, a graph-colouring problem that is motivated by the
question of non-planar, non-revisiting cycles in a polyhedral map, is considered. The
conjecture is shown to be true for all planar graphs and K3j3. Consequently, Bar
nettes result [6] on the existence of the above mentioned cycles on the projective
plane, torus, and Klein bottle is extended to the surfaces V3, and S'2.
3.1 Polygonal Representation of Polyhedral Maps
It is well known that any compact, connected surface may be represented as a
polygon in the plane with labeled and directed sides. The directed sides are matched
35


,?Ui


BIOGRAPHICAL SKETCH
Hari Pulapaka was born in Bombay, India, on March 19, 1966. Upon receiving a
bachelors degree in mathematics from St. Xaviers College, University of Bombay,
he arrived in the U.S. in 1987 to attend graduate school. In 1989, he received an M.S.
in mathematics from George Mason University, Fairfax, VA, under the supervision
of Dr. James Lawrence. And in 1995, he received a Ph.D. in mathematics from the
University of Florida, Gainesville, FL, under the supervision of Dr. Andrew Vince.


46
planar revisits. Hence by Lemma 2.1.1, there is a non-revisiting path joining u and
v in M.
3.2 Polygonal Representation and Non-Revisiting Cycles
It is known that any polyhedral map on the projective plane, torus or Klein bottle
has a non-planar, non-revisiting cycle. Barnettes proofs [6] of these results are not
trivial and involve some details. In this section, we give a unified, elementary proof
of these results in the case where the polyhedral map has a non-separating polygonal
representation. This is done by considering the cycles of M that lie on the boundary
of a non-separating polygonal representation of M. Such cycles are non-planar by
Lemma 3.2.1 below. And by techniques that are similar to those used in Chapter
2, it is shown that if a cycle of M contained in dP revisits a face, then it can be
modified to a cycle that is non-revisiting and is also contained in dP. In the next
section, a graph-colouring conjecture is proposed and it is shown that the conjecture
is true for all planar graphs. Consequently, an alternate proof of the above-stated
result on non-revisiting cycles is given.
Lemma 3.2.1. Let M (G,S) be a polyhedral map on a surface and P be a
polygonal representation of M. If C is a cycle of M that is contained in dP, then it
must be a non-planar cycle in M.
Proof. The proof is by contradiction; so assume that C bounds a cell A in M. Let
/ be a face of M that is not contained in A and that has an edge e in common with C.
Such a face exists; otherwise A would contain all the faces in M which is impossible.
Let e be the matching edge for e on dP and let f be the face of P that contains


44
Pn* has 3?^ 7 or 3n 8 vertices and if Pn* has exactly four faces, then the graph
of Pn has An 10,4n 11, or 4n 12 vertices. Now, by the Euler formula, M* has
v* = n^n~1^ n + 2 (n~3Kn~.4l vertices. Note that as n increases, the number of
vertices of the graph of Pn* grows linearly while v* grows quadratically. In fact, for
n > 11, v* is greater than each of the numbers 2n 6,3n 7,3n 8, An 10, An 11
and 4n 12. Thus if n > 11, the graph of Pn* cannot possibly span the graph of
M* and we need only consider the case where n 7. Let M7 be the polyhedral map
corresponding to the polygonal representation shown in Figure 17a. By statements
(1), (2) and (3) above, P* must be a map on a closed disc with 14 vertices and 3
hexagonal faces. Hence the only possibilities for P7* are as shown in Figure 19.
Figure 19. The two possibilities for a 2-connected, planar
map with 14 vertices and 3 hexagonal faces.
However, it is easily checked that these planar maps are not contained in M?.
In Section 2.2, the polyhedral maps M\ and M2 were counterexamples to the
non-revisiting path conjecture for the surfaces S2 and Ah, respectively. Figure 20a
shows a polygonal representation of Mi while Figure 20b gives a polygonal represen
tation for M2. Observe that both polygonal representations are separating, however,
it is easy to construct similar counterexamples that have non-separating polygonal
representations.


33
for otherwise, either F and F9 meet improperly which is a contradiction since Mi is
a polyhedral map, or G and T meet improperly which is a contradiction since Mi
was chosen to be a polyhedral map. The proof that x and y cannot be joined by
a non-revisiting path in Mi is identical to the proof given earlier. Thus, Mi is a
polyhedral map on Sg without the non-revisting property.
Next, we show that the non-revisiting path conjecture is false for the surface N4.
In this case, consider the 17 polygons Fi,..., Fi7 with the vertex-labelling shown in
Figure 14.
Figure 14. The faces that constitute a counter-example to the
non-revisiting path conjecture for polyhedral maps on iV4.
As in the case of the surface S2, paste the polygons together by identifying edges
with the same labels. Again, the result is a surface S without boundary with the
map M2 given by the union of the faces F\,..., F17. It can be checked that the faces


CHAPTER 4
CONCLUSION
The non-revisiting path conjecture is now settled for all polyhedral maps except
those that are homeomorphic to A3. It is conceivable that the conjecture is true
for polyhedral maps on surfaces homeomorphic to A3, however, at this juncture, no
proof is known. Considering the complexity of the proofs of the validity of the non
revisiting path conjecture for the torus and Klein bottle, a brute force method might
prove to be tedious in the case of the surface A3.
The problem of the existence of non-planar, non-revisiting cycles in a polyhedral
map is wide open The only surfaces for which such cycles are known to exist are the
projective plane, torus, Klein bottle, and for a class of polyhedral maps on the surfaces
homeomorphic to A3 and S2 (see Theorem 3.3.3). If every polyhedral map contains a
non-planar, non-revisiting cycle, one could potentially obtain decomposition theorems
for surfaces other than the torus and Klein bottle.
Even though the graph coloring conjecture proposed in Section 3.3 was motivated
by the existence of non-planar, non-separating cycles in a polyhedral map, the conjec
ture is certainly interesting on its own merit. Apart from planar graphs, graphs with
a triangle, ^3,3, and some very specific graphs (not included in this dissertation), the
coloring conjecture remains unsettled.
In conclusion, this research has raised at least three different questions that remain
unsolved and would make for some interesting work in the future.
60


20
u
Figure 5. The six possibilities for a non-planar revisit of I\
to F\ in the case in Figure 4a.


26
u
Figure 9. The two possibilities fo.r a non-planar revisit of F2
to F2 in the case in Figure 8.
Figure 10. A non-planar revisit of T to F in the case where
both vertices are in the interior of the annulus.


52
Thus in all cases M has a non-planar, non-revisiting cycle that is contained in
dP. m
3.3 A Graph-Coloring Problem and Non-Revisiting Cycles
A path P in a graph G is said to be a chord of a cycle C in G if P is a path joining
vertices x and y of C such that P fl C = {x:y}. For our purposes, an edge-coloring
of G is a coloring of the edges of G in which every edge can be colored with many
different colors. Given such an edge-coloring of G, a subgraph H of G is said to be
monochromatic if there is a color C\ such that every edge of H is colored with C\.
Similarly, H is said to be dichromatic if there are colors Ci and C- such that every
edge of H is colored with Ci or C2.
Conjecture. Every edge-colored finite graph G with no mono or dichromatic cy
cles contains a cycle with no monochromatic chord.
The above conjecture is motivated by the problem of the existence of non-planar,
non-revisiting cycles in a polyhedral map. If every edge is coloured using exactly
two colors, then the validity of the coloring conjecture implies that every polyhedral
map that has a non-separating polygonal representation, in fact, has a non-planar,
non-revisiting cycle. The proof of this result follows.


37
A A
B B B B
Figure 15. An improper matching of edges on dP.
Theorem 3.1.1. Every polyhedral map M, not on the sphere, has a polygonal
representation.
Proof. Label the vertices of M. Since the underlying graph of the dual map M*
is connected, it has a spanning tree T*. There is a bijection between the edges e in
E(M) and the edges e* in E(M*). Here e* is the unique edge that crosses e. Let E*
denote the complement of T* in M* and define E = {e 6 E(M)|e* 6 E*}. Cut M
along the edges in E. Since T* is planar, the result is a planar map P that satisfies
all the conditions for it to be a polygonal map except condition (1). If a pair of edges
are matched as in Figure 15, then glue them back together. Now every vertex label
on dP appears at least twice on dP and the map still remains planar. Furthermore,
P is a map that satisfies all the conditions for it to be a polygonal representation of
M. u
Let P be a polygonal map and assume that a pair of directed edges (A, B) and
(i?, C) on dP are incident at B. Further assume that the respective matching edges
(A, By and (B, C)' are also incident at B. Replace (A, B) and (P, C) by a single
directed edge (A, (7); similarly replace (A, B)' and (P, C)' by a single directed edge
(A, C)'. Call such a replacement a concatenation. Perform concatenations along dP
until it is no longer possible to do so. Call dP with the resulting vertex labeling
the type of P, denoted by Tp. Figure 16 shows a polyhedral map M on the torus, a
polygonal representation of M, and the type of P.


15
2. u lies on C and v does not lie on C. Since every vertex of M has degree at least
three, there must be a vertex ui of M in the interior of the cell H such that uui is an
edge of M. Since the cycle C is non-revisiting, the intersection of any face of M with
dH is either empty, or connected. Hence by Lemma 2.1.2, there is a path Tq joining
iii and v in M that is contained in the interior of H. Define T = r0 U uui. Thus
T is a path joining u and v that meets the boundary of H in only u. If T has only
planar revisits, we are done by Lemma 2.1.1. It is clear that a non-planar revisit of
T to a face Fmust involve a vertex s lying on r(u,u] and u. Among all non-planar
revisits of T, choose F so that s is nearest to v along T. Replace T by the path
Ti F[u,s] U r[s,u] indicated by the dotted path in Figure 2.
t
Figure 2. A non-planar revisit of T to F on the projective plane.


59
by the same argument given for the surface _/V3, there is a non-planar, non-revisiting
cycle in M that is contained in dP. m


CHAPTER 2
NON-REVISITING PATHS ON SURFACES
This chapter deals with the non-revisiting path conjecture for polyhedral maps. In
Section 2.1, the non-revisiting path conjecture is shown to be true for polyhedral maps
on the projective plane, torus, and Klein bottle. Although these results are already
known, the earlier proofs due to Barnette [5,7] and Engelhardt [10], are different. In
Section 2.1, a simpler, unified proof for all three surfaces is provided. In the case of
the projective plane, by a result due to Barnette [6], the surface is cut along a non-
planar, non-revisiting cycle to yield a cell and the proof consists of the considering
the following three cases:
1. Both vertices involved lie on the boundary of the cell.
2. One vertex lies on the boundary of the cell while the other lies in the interior of
the cell.
3. Both vertices lie in the interior of the cell.
In the case of the torus and Klein bottle, the surface is cut along a non-revisiting
cycle in the polyhedral map yielding an annulus. The unification of the proof for all
three surfaces is achieved by considering the same three cases stated above.
In Section 2.2, the non-revisiting path conjecture is settled for polyhedral maps
on all the remaining surfaces except A3, the connected sum of three copies of the
projective plane (or equivalently, the connected sum of the torus and the projective
plane, or the Klein bottle and the projective plane). Specifically, it is shown that the
non-revisiting path conjecture is false for all the remaining surfaces, except possibly
A3 and counter-examples are provided to this effect.
10


34
of S cannot be assigned an orientation as described earlier in the proof. To check if
such an assignment is possible, first assign an arbitrary orientation to a particular
face. This forces an orientation of each face that shares a common edge with the
original face. Since the surface is connected, the process can be continued untill the
orientation of each face has been forced. Either the result is an orientation for the
embedding, or else the given embedding has no orientation in which case the surface
is non-orientable. Hence S is non-orientable. M2 has 11 vertices, 30 edges, and 17
faces. Thus S has Euler Characteristic 2 and must be homeomorphic to jV4. Once
again, it can be checked that the faces F\,.... Fn meet properly. Hence M2 is a
polyhedral map on IV4. The proof that M does not have the non-revisiting property
is identical to the one given for the surface 62.
A counterexample for the surface Nk > 5 is obtained by glueing a polyhedral
map on A^_4 to M2 as described in the orientable case. Again, this method yields a
counterexample for each surface N^, k > 5.