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Counterexamples ( jstor ) Genetic mapping ( jstor ) Geometric planes ( jstor ) Klein bottles ( jstor ) Mathematics ( jstor ) Polygons ( jstor ) Polyhedrons ( jstor ) Polytopes ( jstor ) Vertices ( jstor )
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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. 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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 |

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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 nÂ¡ > 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 vÂ¡ 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 vÂ¡ 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 faceF2 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 tÂ¡ 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. 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'RFWRU RI 3KLORVRSK\ PRVRSQ\ -7OXVUn %HUQDUG 0DLU $VVRFLDWH 3URIHVVRU RI 0DWKHPDWLFV FHUWLI\ WKDW KDYH UHDG WKLV VWXG\ DQG WKDW LQ P\ RSLQLRQ LW FRQIRUPV WR DFFHSWn DEOH VWDQGDUGV RI VFKRODUO\ SUHVHQWDWLRQ DQG LV IXOO\ DGHTXDWH LQ VFRSH DQG TXDOLW\ DV D GLVVHUWDWLRQ IRU WKH GHJUHH RI 'RFWRU RI 3KLORVA 1HLO /ZKLWH 3URIHVVRU RI 0DWKHPDWLFV 7KLV GLVVHUWDWLRQ ZDV VXEPLWWHG WR WKH *UDGXDWH )DFXOW\ RI WKH 'HSDUWPHQW RI 0DWKHPDWLFV LQ WKH &ROOHJH RI /LEHUDO $UWV DQG 6FLHQFHV DQG WR WKH *UDGXDWH 6FKRRO DQG ZDV DFFHSWHG DV SDUWLDO IXOILOOPHQW RI WKH UHTXLUHPHQWV IRU WKH GHJUHH RI 'RFWRU RI 3KLORVRSK\ $XJXVW 'HDQ *UDGXDWH 6FKRRO 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 (Â£?, 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^rhÂ¡. 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. |