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Cliques, Degrees, and Coloring: Expanding the Ω, ∆, Χ Paradigm

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Cliques, Degrees, and Coloring: Expanding the Ω, ∆, Χ Paradigm Cliques, Degrees, and Coloring: Expanding the !; ∆; χ paradigm by Thomas Kelly A thesis presented to the University of Waterloo in fulfillment of the thesis requirement for the degree of Doctor of Philosophy in Combinatorics & Optimization Waterloo, Ontario, Canada, 2019 c Thomas Kelly 2019 Examining Committee Membership The following served on the Examining Committee for this thesis. The decision of the Examining Committee is by majority vote. External Examiner: Alexandr Kostochka Professor, Dept. of Mathematics, University of Illinois at Urbana-Champaign. Supervisor: Luke Postle Associate Professor, Dept. of Combinatorics & Optimization, University of Waterloo. Internal Members: Penny Haxell Professor, Dept. of Combinatorics & Optimization, University of Waterloo. Jim Geelen Professor, Dept. of Combinatorics & Optimization, University of Waterloo. Internal-External Member: Lap Chi Lau Associate Professor, Cheriton School of Computer Science, University of Waterloo. ii 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. iii Abstract Many of the most celebrated and influential results in graph coloring, such as Brooks' Theorem and Vizing's Theorem, relate a graph's chromatic number to its clique number or maximum degree. Currently, several of the most important and enticing open problems in coloring, such as Reed's !; ∆; χ Conjecture, follow this theme. This thesis both broadens and deepens this classical paradigm. In PartI, we tackle list-coloring problems in which the number of colors available to each vertex v depends on its degree, denoted d(v), and the size of the largest clique containing it, denoted !(v). We make extensive use of the probabilistic method in this part. We conjecture the \list-local version" of Reed's Conjecture, that is every graph is L- colorable if L is a list-assignment such that jL(v)j ≥ d(1 − ")(d(v) + 1) + "!(v))e for each vertex v and " ≤ 1=2, and we prove this for " ≤ 1=330 under some mild additional assumptions. We also conjecture the \mad version" of Reed's Conjecture, even for list-coloring. That is, for " ≤ 1=2, every graph G satisfies χ`(G) ≤ d(1 − ")(mad(G) + 1) + "!(G)e; where mad(G) is the maximum average degree of G. We prove this conjecture for small values of ", assuming !(G) ≤ mad(G) − log10 mad(G). We actually prove a stronger result that improves bounds on the density of critical graphs without large cliques, a long-standing problem, answering a question of Kostochka and Yancey. In the proof, we use a novel application of the discharging method to find a set of vertices for which any precoloring can be extended to the remainder of the graph using the probabilistic method. Our result also makes progress towards Hadwiger's Conjecture: we improve the best known bound on the chromatic number of Kt-minor free graphs by a constant factor. We provide a unified treatment of coloring graphs with small clique number. We prove that for ∆ sufficiently large, if G is a graph of maximum degree at most ∆ with list- assignment L such that for each vertex v 2 V (G), (s ) ln(!(v)) !(v) ln(ln(d(v))) log (χ(G[N(v)]) + 1) jL(v)j ≥ 72 · d(v) min ; ; 2 ln(d(v)) ln(d(v)) ln(d(v)) iv and d(v) ≥ ln2 ∆, then G is L-colorable. This result simultaneously implies three famous results of Johansson from the 90s, as well as the following new bound on the chromatic number of any graph G with !(G) ≤ ! and ∆(G) ≤ ∆ for ∆ sufficiently large: r ln ! χ(G) ≤ 72∆ : ln ∆ In Part II, we introduce and develop the theory of fractional coloring with local de- mands. A fractional coloring of a graph is an assignment of measurable subsets of the [0; 1]-interval to each vertex such that adjacent vertices receive disjoint sets, and we think of vertices \demanding" to receive a set of color of comparatively large measure. We prove and conjecture \local demands versions" of various well-known coloring results in the !; ∆; χ paradigm, including Vizing's Theorem and Molloy's recent breakthrough bound on the chromatic number of triangle-free graphs. The highlight of this part is the \local demands version" of Brooks' Theorem. Namely, we prove that if G is a graph and f : V (G) ! [0; 1] such that every clique K in G satisfies P v2K f(v) ≤ 1 and every vertex v 2 V (G) demands f(v) ≤ 1=(d(v) + 1=2), then G has a fractional coloring φ in which the measure of φ(v) for each vertex v 2 V (G) is at least f(v). This result generalizes the Caro-Wei Theorem and improves its bound on the independence number, and it is tight for the 5-cycle. v Acknowledgements Most of all, I would like to thank my advisor, Luke Postle, for his guidance over the last four years. I am grateful to have learned so much from his direction and insight. I would also like to thank Luke for encouraging me to travel, which has broadened my studies considerably. On that note, I would like to thank everyone that I have visited and with whom I have collaborated over the last few years. I would especially like to thank Marthe Bonamy and Peter Nelson for all of the productive visits as part of the Bordeaux-Waterloo project, which led to the \Low Omega" paper, an integral component of this thesis. I would also like to thank Alexandr Kostochka, Penny Haxell, Jim Geelen, and Lap Chi Lau for serving on my committee. Lastly, I want to thank my friends and family. In particular, I want to thank all of the people that have made living in Waterloo fun, especially the Dawson house, and Jane. vi Table of Contents List of Tables xii List of Figures xiii 1 Introduction1 1.1 Graph coloring preliminaries..........................2 1.1.1 Graph theory terminology.......................2 1.1.2 List coloring...............................4 1.1.3 Fractional coloring...........................6 1.2 !; ∆; and χ ...................................8 1.2.1 Brooks' Theorem............................9 1.2.2 Reed's Conjecture............................9 1.2.3 Coloring graphs with small clique number.............. 12 1.3 Local list versions................................ 14 1.3.1 A local version of Reed's Conjecture................. 15 1.3.2 A unifying local version........................ 17 1.4 Critical graphs................................. 19 1.4.1 An application to Hadwiger's Conjecture............... 21 1.5 Fractional coloring with local demands.................... 23 1.5.1 Local demands version of Brooks' Theorem.............. 23 1.5.2 Edge-coloring and χ-boundedness................... 26 1.5.3 Small clique number.......................... 27 vii 2 Probabilistic method preliminaries 29 2.1 The basics.................................... 30 2.2 Our toolkit................................... 30 2.2.1 The Lov´aszLocal Lemma....................... 31 2.2.2 Basic concentration inequalities.................... 32 2.3 A new concentration inequality........................ 34 I List coloring 40 3 The local naive random coloring procedure 42 3.1 A metatheorem................................. 44 3.2 Proof of Theorem 3.1.8............................. 48 3.3 Concentrations................................. 50 3.3.1 Exceptional outcomes.......................... 50 3.3.2 Proof of Lemma 3.2.3.......................... 52 3.4 Sparsity..................................... 57 3.4.1 Improving the constants........................ 57 3.4.2 Local sparsity, cascading, and more potential applications...... 60 4 The local version of Reed's Conjecture 62 4.1 Local sparsity.................................. 63 4.2 Density..................................... 66 4.2.1 A density lemma............................ 67 4.2.2 The egalitarian neighborhood is sparse................ 70 4.3 Proof of Theorem 1.3.4............................. 73 viii 5 The density of critical graphs with no large cliques 76 5.1 Applying Theorem 1.3.4 when ! < k=2.................... 76 5.2 Overview of the proof of Theorem 5.0.1.................... 78 5.2.1 The savings............................... 79 5.2.2 The dense vertices........................... 80 5.2.3 The discharging............................. 80 5.3 Coloring a saved graph with Theorem 3.1.8.................. 81 5.3.1 Ways to save.............................. 81 5.3.2 Expectations.............................. 83 5.3.3 Applying Theorem 3.1.8........................ 87 5.4 Finding a saved subgraph........................... 88 5.4.1 A stronger version........................... 88 5.4.2 Preliminaries.............................. 90 5.4.3 Structure................................ 92 5.4.4 Discharging............................... 99 5.4.5 Proof of Lemma 5.4.1.......................... 103 5.5 Putting it all together............................. 103 6 Coloring graphs with small clique number 106 6.1 More background................................ 106 6.1.1 The nibble method and the finishing blow.............. 107 6.1.2 A more general Theorem........................ 109 6.2 Proof of Theorem 6.1.2............................. 110 6.2.1 Partial colorings............................. 111 6.2.2 Completing a partial coloring..................... 111 6.2.3 Analyzing a random partial coloring................. 112 6.2.4 Finding the partial coloring...................... 115 6.3 Proofs of Theorems 1.3.7 and 1.3.11...................... 116 ix 6.3.1 Bounding the average size of an independent set........... 116 6.3.2 The proofs................................ 118 6.4 Proof of Theorem 1.3.12...........................
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