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Colourings of Graphs and Words Research Collection Doctoral Thesis Colourings of Graphs and Words Author(s): Kamčev, Nina Publication Date: 2018 Permanent Link: https://doi.org/10.3929/ethz-b-000282692 Rights / License: In Copyright - Non-Commercial Use Permitted This page was generated automatically upon download from the ETH Zurich Research Collection. For more information please consult the Terms of use. ETH Library COLOURINGS OF GRAPHS AND WORDS DISS. ETH No. 25147 Nina Kamčev nina kamcevˇ COLOURINGSOFGRAPHSANDWORDS diss. eth no. 25147 COLOURINGSOFGRAPHSANDWORDS A dissertation submitted to attain the degree of doctor of sciences of eth zurich (Dr. sc. ETH Zurich) presented by nina kamcevˇ MMath (Cantab) born on 7 March 1991 citizen of Croatia accepted on the recommendation of Prof. Dr. B. Sudakov, examiner Prof. Dr. M. Axenovich, co-examiner 2018 Nina Kamˇcev: Colourings of Graphs and Words, © 2018 doi: To my family and my Father. soli deo gloria ABSTRACT Extremal graph theory is concerned with the extreme values of a graph parameter over various classes of graphs. Randomised constructions have played a major role in extremal combinatorics. This phenomenon acted as a catalyst for the development of probabilistic combinatorics and the theory of random graphs as independent research areas. In the present thesis, we consider three graph parameters – anagram- chromatic number, rainbow connectivity and zero forcing number. Each of them is lower-bounded by a corresponding well-studied parameter (the Thue-chromatic number, diameter and minimum rank respectively). Our aim is to understand the hierarchy of graphs delineated by the parameter in consideration, and to highlight the role of random graphs as a surprising or close-to-optimal examples. Furthermore, we analyse a random graph process constrained to the König property. A graph is called König if the size of its maximum match- ing is equal to the order of its minimal vertex cover. We answer questions about the evolution and the final outcome of the process. A common feature of our proofs is that parts of them parallel classical problems concerning existence of relevant substructures in random graphs (Hamilton cycle, spanning tree, independent set, perfect matching). This allows us to rely on some well-known approaches. The second part of the thesis contains two Ramsey-type results. A catch- phrase of Ramsey theory is that ‘any sufficiently large structure contains a well-organised substructure’. Let c be an edge-colouring of the complete n-vertex graph Kn. The prob- lem of finding properly coloured and rainbow Hamilton cycles in c was initiated in 1976 by Bollobás and Erd˝osand has been extensively studied since then. We study the problem in a more general hypergraph setting, giving sufficient local (resp. global) restrictions on the colourings which guarantee a properly coloured (resp. rainbow) copy of a given hypergraph G. We also look at multipartite analogues of these questions. In our final chapter, the ‘well-organised substructure’ of interest is a combinatorial line in [m]n. The Hales–Jewett theorem states that for any m and r there exists an n such that any r-colouring of the elements of [m]n contains a monochromatic combinatorial line. We look more closely at the ix obtained combinatorial line and prove tight results about the structure of its active coordinate set. x ZUSAMMENFASSUNG Extremale Graphentheorie befasst sich mit den Extremwerten eines Gra- phenparameters über verschiedene Klassen von Graphen. Randomisierte Konstruktionen spielen eine wichtige Rolle in der Extremalkombinatorik. Dieses Phänomen wirkte als Katalysator für die Entwicklung der probabi- listischen Kombinatorik und der Theorie der Zufallsgraphen als unabhän- gige Forschungsgebiete. In der vorliegenden Arbeit betrachten wir drei Graphenparameter – die Anagramm-färbungszahl, den Regenbogendurchmesser und die Null- Forcingzahl. Jeder von ihnen hat einen entsprechenden, bereits gut un- tersuchten Parameter (die Thue-Färbungszahl, den Durchmesser und den minimalen Rang) als untere Schranke. Unser Ziel ist es, die Hierarchie der Graphen zu verstehen, die durch den betrachteten Parameter beschrieben wird, und die Rolle der Zufallsgraphen als überraschende oder nahezu optimale Beispiele hervorzuheben. Darüber hinaus analysieren wir einen Zufallsgraphprozess, der auf die König-Eigenschaft beschränkt ist. Ein Graph wird König-Graph genannt, wenn die Grösse seines maximalen Matchings der Kardinalität seiner mi- nimalen Knotenüberdeckung entspricht. Wir beantworten Fragen über die Entwicklung und das Endergebnis des Prozesses. Unsere Beweise haben gemein, dass Teile davon Varianten der klassi- schen Probleme bezüglich der Existenz relevanter Unterstrukturen in zu- fälligen Graphen (Hamiltonkreis, Spannbaum, Co-Clique, perfektes Mat- ching) sind. Dies ermöglicht uns, auf einige bekannte Ansätze zurückzu- greifen. Der zweite Teil der Arbeit enthält zwei Ramseytheoretische Ergebnisse. Ein Schlagwort der Ramseytheorie ist, dass "jedes ausreichend grosse Sys- tem ein gut strukturiertes Untersystem enthält". Sei c eine Kantenfärbung der n-Clique Kn. Das Problem, gültig gefärbte und Regenbogen-Hamiltonkreise in c zu finden, wurde in 1976 von Bol- lobás und Erd˝osinitiiert und seither umfassend untersucht. Wir untersu- chen das Problem in einem allgemeineren Hypergraph-Kontext. Wir fin- den hinreichende lokale (bzw. globale) Bedingungen für die Färbungen, die eine gültig gefärbte (bzw. Regenbogen-) Kopie eines gegebenen Hyper- graphen G garantieren. Darüber hinaus werden analoge Fragen für multi- partite Graphen besprochen. xi Im letzten Kapitel interessieren wir uns für eine kombinatorische Gerade in [m]n als "gut strukturiertes Untersystem". Der Satz von Hales–Jewett be- sagt, dass für jedes m und jedes r ein n existiert, so dass jede r -Färbung der Elemente von [m]n eine einfarbige kombinatorische Gerade enthält. Wir be- trachten die erhaltene kombinatorische Gerade genauer und erhalten opti- male Ergebnisse über die Struktur der aktiven Koordinatenmenge. xii ACKNOWLEDGEMENTS My family and my granparents have provided unfailing support through- out my education and time abroad. I would like to thank Benny Sudakov for the opportunities and guidance during my doctoral studies. My PhD siblings and colleagues – Dániel Korándi, Pedro Vieira, Matthew Kwan, Roman Glebov, Jan Volec, Igor Balla, Felix Dräxler, Alexey Pokrovskiy, Jon Noel, Matija Buci´c,Shoham Letzter, Tuan Tran, Domagoj Cevid,´ Seraina Wachter, Ana Canas da Silva, Meike Akveld, Cornelia Busch, Christa Lach- muth, Maria Hempel, Zur Luria, Frank Mousset, Rajko Nenadov, Mohsen Ghaffari, Alex Puttick and a number of others – have made my time at ETH enjoyable. A number of people from the combinatorics community have inspired, encouraged and supported me in my research, including László Babai, Béla Bollobás, David Conlon, Jacob Fox, Stefanie Gerke, Thomas Kalinowski, Michael Krivelevich, Imre Leader, Anita Liebenau, Tomasz Łuczak, Natasha Morrison, Christoph Spiegel, Simone Severini, Oriol Serra, Goran Žuži´c. The template was created by Tino Wagner, saving me a lot of time and frustration. Last, but not least, I have always been able to rely on my friends – Han- nah and Sven Eggimann, Sarah Gales, Zofia Jackson, Samuel Leder, Han- nelore Leder, Steve Matsumoto, Sophia Michael, Yining Nie, Eva Waldis- pühl, Remi Tobler, the iCafé Zurich group. This list is bound to be incom- plete. I extend my gratitude to those omitted. xiii CONTENTS 1 introduction1 2 anagram-free graph colourings9 2.1 Specific families of graphs . 11 2.2 Bounded-degree graphs . 14 3 rainbow connectivity 33 3.1 Edge rainbow connectivity . 35 3.2 Vertex rainbow connectivity . 40 4 zero forcing number 45 4.1 Graphs with forbidden subgraphs . 49 4.2 The random graph . 52 4.3 Spectral bounds . 67 5 the könig graph process 73 5.1 Overview of Proof . 76 5.2 Preliminaries and Probabilistic Tools . 79 5.3 Forming a large matching . 83 5.4 The structure of GN ......................... 88 5.5 Rigidity and uniqueness of an optimal cover . 91 5.6 Delayed perfect matching threshold . 94 5.7 Open Questions . 109 6 bounded colourings of multipartite graphs and hy- pergraphs 111 6.1 Lovász local lemma and the Lu-Székely framework for ran- dom injections . 116 6.2 Embedding m-partite graphs . 117 6.3 Embedding bounded-degree hypergraphs . 123 7 intervals in the hales–jewett theorem 131 a multidimensional lu-székely 135 bibliography 139 xv NOTATION frequently used symbols symbol meaning N natural numbers [n] the set f1, 2, . , ng S(k) family of k-order subsets of S log logarithm base e V(G) vertex set of G E(G) edge set of G d(G) minimum degree D(G) maximum degree degG(v) degree of v NG(v) neighbours of v (excluding v) S NG(S) ( v2S NG(v)) n S G[S] the subgraph of G induced on S u ∼ v u and v are adjacent EG(S) edges of G with both endpoints in S eG(S) jEG(S)j EG(S, T) edges of G with one endpoint in S and one in T eG(S, T) jEG(S, T)j (r) Kn the complete r-uniform graph on n-vertices; r = 2 if sup- pressed Ka,b the complete bipartite graph with parts of order a and b Cn the n-vertex cycle G(n, m) random graph with m edges G(n, p) binomial random graph with edge probability p Gn,d random d-regular graph xvi 1 INTRODUCTION Extremal graph theory in its broad sense studies the relations between dif- ferent graph properties, often with the objective of maximising a specific graph parameter over an appropriate class of graphs. Properties which are often considered are connectivity, (minimum, maximum or average) degree, clique number, chromatic number, diameter, Hamiltonicity etc. A common theme across several chapters of this thesis is the crucial role played by random graphs and related randomised constructions in ex- tremal problems. Hence we start by reviewing some classical results in this direction, a brief history of random graph models and our contribu- tion to the developments (Chapters 2-5). Then we introduce our final two results and put them in the context of Ramsey theory. Many terms in the introduction are left undefined to retain the flow.
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