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Extensions of Galvin's Theorem Extensions of Galvin's Theorem by Maxwell Levit A thesis presented to the University of Waterloo in fulfillment of the thesis requirement for the degree of Masters of Mathematics in Combinatorics & Optimization Waterloo, Ontario, Canada, 2018 c Maxwell Levit 2018 I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. ii Abstract We discuss problems in list coloring with an emphasis on techniques that utilize oriented graphs. Our central theme is Galvin's resolution of the Dinitz problem (Galvin. J. Comb. Theory, Ser. B 63(1), 1995, 153{158). We survey the related work of Alon and Tarsi (Combinatorica 12(2) 1992, 125{134) and H¨aggkvistand Janssen (Combinatorics, Probability & Computing 6(3) 1997, 295{313). We then prove two new extensions of Galvin's theorem. iii Acknowledgements I would like to thank my supervisor, Joseph Cheriyan, and my readers, Penny Haxell and Luke Postle. iv Table of Contents List of Figures vii 1 Introduction 1 2 The Polynomial Method 4 2.1 Definitions and Notations . .4 2.2 The Graph Polynomial . .5 2.3 Oriented Graphs . .6 2.4 The Alon-Tarsi theorem . .9 2.5 Applications to list coloring . 10 2.6 The Alon-Tarsi conjecture . 12 2.7 The Combinatorial Nullstellensatz . 14 3 A result of H¨aggkvistand Janssen 15 3.1 The list chromatic index of the complete graph . 15 3.2 A bound for simple graphs . 30 4 Extensions of Galvin's Theorem 35 4.1 Preliminaries . 36 4.1.1 Coloring . 36 4.1.2 Stable Matchings . 37 4.1.3 Oriented graphs . 37 4.1.4 A convenient correspondence . 39 4.1.5 Preference maps and completions . 40 4.1.6 Galvin's Theorem . 43 4.1.7 The Hungarian Forest . 43 4.2 Non-uniqueness of Galvin colorings . 45 4.3 Weakening the hypothesis . 49 4.4 An application to minimal listings . 54 5 The List Chromatic Number of the Hamming Graph 57 5.1 A Geometric Viewpoint . 57 5.2 Lower bounds . 58 v 5.3 Upper bounds . 60 6 Conjectures and Open Problems 63 6.1 Further extension of Galvin's theorem . 63 6.1.1 Enumeration of list colorings . 63 6.1.2 A coarser equivalence . 65 6.1.3 Compression of lists . 65 6.2 k-sheets . 67 6.3 The list chromatic number of the hypercube . 68 6.3.1 A simple upper bound . 68 6.3.2 An asymptotic lower bound in d ..................... 68 References 72 vi List of Figures 2.1 A planar bipartite graph exhibiting an unsatisfiable 2-listing. 11 3.1 Diagram for Lemma 14 . 16 3.2 K4 and its line graph. 21 3.3 Illustration of the algorithm provided in the proof of Theorem 17 . 23 3.4 After the first out degree is assigned . 23 3.5 After the second out degree is assigned . 24 3.6 After the third out degree is assigned . 24 3.7 After the fourth out degree is assigned . 25 3.8 After the fifth out degree is assigned . 25 3.9 The completed orientation . 26 3.10 Illustration of the order in which vertices of K9 are assigned out degrees . 28 3.11 Illustration of Theorem 22 . 33 4.1 A bipartite graph G and preference system P ................. 38 4.2 The orientation Λ(G; P ) of the preference graph (G; P ) shown in Figure 4.1 . 39 4.3 A preference map Φ on K3;3 and the preference system [Φ] . 40 4.4 Completable and incompletable partial preference maps . 41 4.5 The two cases from Lemma 28 . 42 4.6 The Hungarian forests rooted at A0 and B0 and the induced M-partition . 44 4.7 Illustration for Lemma 32 . 45 4.8 Illustration for Theorem 34 . 48 4.9 Two copies of Λ(K3;3), equipped with unsatisfiable listings . 49 4.10 Illustration of a key observation . 50 4.11 Illustration for Lemma 38 . 52 4.12 Second illustration for Lemma 38 . 53 5.1 The listing described in Lemma 42 . 59 5.2 Illustration of the Hypergraph ξ(4; 3)...................... 62 6.1 A listing for which Galvin's theorem finds few L-colorings . 64 6.2 A 3-listing and its compression . 65 6.3 Illustration for Fact 4 . 66 6.4 A graph exhibiting an unsatisfiable 2-listing . 69 vii 6.5 A subgraph of Q63 consisting of a cube and all of its neighbors . 70 6.6 A subgraph of Q63 exhibiting the unsatisfiable listing described in Fact 6 . 71 viii Chapter 1 Introduction It got started when we tried to solve Jeff Dinitz's problem. Erd¨osRubin and Taylor Choosability in Graphs In 1976, Vizing gave the first published account of list coloring in a Russian journal. His work had not reached Erd¨os,Rubin, and Taylor by 1979, when the trio introduced the notion of list coloring to the English speaking research community. Their paper, \Choosability in Graphs" incited research on numerous problems in list coloring. See [42], [15]. Probably the most infamous such problem is the so called edge list coloring conjecture. This conjecture asserts that the chromatic number and list chromatic number coincide for line graphs. Over the course of the 1980's, Bollob´asand Harris, Chetwynd and H¨aggkvist,and Bollob´asand Hind, used probabilistic methods to give successively stronger upper bounds on the list chromatic number of certain very general families of line graphs, see [6], [10], [7]. Following these probabilistic advances, Alon and Tarsi used algebraic techniques to prove that planar bipartite graphs are 3-choosable. In so doing, they codified a powerful polynomial method for proving upper bounds on the list chromatic number. Janssen employed the methods of Alon and Tarsi to show that the edge list coloring conjecture is true for the line graphs of complete bipartite graphs provided that the parts are of unequal size, see [3], [23]. Finally, in 1995, Galvin gave a very surprising proof of the edge list coloring conjecture for the line graph of any bipartite graph. This improved upon Jansen's result, and resolved the so called Dinitz Problem, which had served as an impetus for Erd¨os,Rubin, and Taylor so many years before. Galvin's proof is neither probabilistic, nor algebraic. Rather, it employs an unpublished lemma of Bondy, Boppana, and Siegel, which gives an upper bound on the list chromatic 1 number under a very special structural assumption of the underlying graph. What is remark- able about Galvin's theorem is that this special structural assumption is precisely guaranteed by the famed Gale-Shapely theorem of the 1960's, see [17], [18]. The contribution of this thesis is to prove new generalizations of Galvin's theorem. We begin with two expository chapters. In Chapter 2 we discuss the algebraic approach to list coloring pioneered by Alon and Tarsi. We also discuss the application of their technique to the Dinitz problem. In Chapter 3 we discuss a further application of Alon and Tarsi's methods due to H¨aggkvistand Janssen, see [19]. This work expands upon Janssen's techniques in [23], in order to prove that the list coloring conjecture is true for complete graphs on an odd number of vertices. In Chapter 4 our main technical contribution is Lemma 33 (page 46), wherein we utilize the matching theory of bipartite graphs to construct special preference systems and stable matchings. This lemma will allow us to prove two new extensions of Galvin's theorem. Theorem 34 (page 47) shows that the coloring guaranteed by Galvin's theorem is not unique by finding two distinct list colorings, and Theorem 35 (page 49) finds a list coloring for a more restrictive type of listing. In Chapter 5 we introduce a natural generalization of the Dinitz problem by considering the list chromatic number of the Hamming graph H(n; d). To our knowledge, this problem has not previously been studied. In Lemma 44 (page 60) we give a non-trivial upper bound of 2n − 1 ≥ ch(H(n; 3)) and in Theorem 43 (page 60) we give a non-trivial lower bound of ch(H(n; d)) ≥ n + 1 for all d ≥ 3. We conclude the chapter with Theorem 41 (page 57), where we apply Kahn's result from [24] to prove that for fixed d, ch(H(n; d)) = n + o(n). In Chapter 6 we discuss the obstacles in improving the results of Chapters 4 and 5. We formulate several related conjectures and open questions. Before proceeding to our main content, we conclude our survey of related literature. The proof of Galvin's theorem makes use of an oriented digraph which is morally equivalent to a latin square. For our extensions, we will touch upon the theory of latin squares, in particular, the theory of the completions of partial latin squares. M. Hall initiated this theory by proving that every latin rectangle is completable to a latin square, see [20]. The statement analogous to M. Hall's theorem in higher dimensions is false. Kochol showed this by constructing 3-dimensional examples of non-completable latin cuboids, see [26]. More recently Bryant et al. have generalized this work, see [9]. Although there is no logical dependence between their work and ours, the ideas in Chapter 5 are of a similar flavor. Evans conjectured that any partial latin square with at most n − 1 entries can be completed, see [16].
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