Charles University in Prague Faculty of Mathematics and Physics DOCTORAL THESIS Bernard Lidick´y Graph coloring problems Department of Applied Mathematics Supervisor of the doctoral thesis: doc. RNDr. Jiˇr´ıFiala, Ph.D. Study programme: Computer Science Specialization: Discrete Models and Algorithms Prague 2011 Vˇsemlak´yrn´ık˚um At the first place, I would like to express many thanks to my supervisor Jiˇr´ı Fiala for his great attitude, patience, support and fruitful discussions during my studies. I also want to thank people from Department of Applied Mathematics at Charles University for creating an excellent working environment. In particular to Zdenˇek Dvoˇr´ak,Daniel Kr´al',Jan Kynˇcland OndˇrejPangr´acfor working with me on some topics covered by this thesis. Great thanks also belong to my co-workers from other universities for spending their time working with me on problems solved in this thesis. Namely to Jan Ekstein, Rok Erman, Fr´ed´ericHavet, Pˇremysl Holub, Ken-ichi Kawarabayashi, Sandi Klavˇzarand Riste Skrekovski.ˇ 3 I declare that I carried out this doctoral thesis independently, and only with the cited sources, literature and other professional sources. I understand that my work relates to the rights and obligations under the Act No. 121/2000 Coll., the Copyright Act, as amended, in particular the fact that the Charles University in Prague has the right to conclude a license agreement on the use of this work as a school work pursuant to Section 60 paragraph 1 of the Copyright Act. In ............. date ............ signature 4 N´azevpr´ace:Varianty probl´emu obarven´ı Autor: Bernard Lidick´y Katedra: Katedra aplikovan´ematematiky Vedouc´ıdisertaˇcn´ıpr´ace:doc. RNDr. Jiˇr´ıFiala, Ph.D. Abstrakt: V t´etopr´acistudujeme barven´ıgraf˚u. Pr´aceje rozdˇelenado tˇr´ıˇc´ast´ı, kde kaˇzd´a ˇc´astse zab´yv´ajinou variantou barven´ı. V prvn´ıˇc´astise zab´yv´ame6-kritick´ymigrafy na ploch´ach a 6-kritick´ymigrafy s mal´ympoˇctemkˇr´ıˇzen´ı. Hlavn´ımi v´ysledkyjsou kompletn´ıseznam 6-kritick´ych graf˚una Kleinovˇel´ahvi a 6-kritick´ych graf˚us kˇr´ıˇz´ıc´ım ˇc´ıslemnejv´yˇseˇctyˇri. Druh´aˇc´ast je vˇenov´anavyb´ıravosti rovinn´ych graf˚ubez kr´atk´ych cykl˚u.Ukazujeme, ˇzerovinn´egrafy bez 3-,7- a 8-cykl˚ujsou 3-vyb´ırav´ea ˇzerovinn´egrafy bez troj´uheln´ık˚u a s jist´ymomezen´ımna 4-cykly jsou t´eˇz3-vyb´ırav´e. V posledn´ıˇc´astise zamˇeˇrujemena novˇejˇs´ıvariantu barven´ı- vlnov´ebarven´ı. Jde o koncept, kter´yje motivovan´ypˇriˇrazov´an´ımfrekvenc´ıa m´azohlednit, ˇzer˚uzn´efrekvence maj´ır˚uzn´ydosah. Zab´yv´amese pak zlepˇsov´an´ımodhad˚unutn´ehopoˇctubarev k obar- ven´ıˇctvercov´ea ˇsesti´uheln´ıkov´emˇr´ıˇzky. Kl´ıˇcov´aslova: kritick´egrafy, seznamov´ebarven´ı,vlnov´ebarven´ı,rovinn´egrafy, kr´atk´ecykly Title: Graph coloring problems Author: Bernard Lidick´y Department: Department of Applied Mathematics Supervisor: doc. RNDr. Jiˇr´ıFiala, Ph.D. Abstract: As the title suggests, the central topic of this thesis is graph coloring. The thesis is divided into three parts where each part focuses on a different kind of coloring. The first part is about 6-critical graphs on surfaces and 6-critical graphs with small crossing number. We give a complete list of all 6-critical graphs on the Klein bottle and complete list of all 6-critical graphs with crossing number at most four. The second part is devoted to list coloring of planar graphs without short cycles. We give a proof that planar graphs without 3-,6-, and 7- cycles are 3-choosable and that planar graphs without triangles and some constraints on 4-cycles are also 3-choosable. In the last part, we focus on a recent concept called packing coloring. It is motivated by a frequency assignment problem where some frequencies must be used more sparsely that others. We improve bounds on the packing chromatic number of the infinite square and hexagonal lattices. Keywords: critical graphs, list coloring, packing coloring, planar graphs, short cycles 5 Contents 1 Introduction 9 2 Definitions 11 2.1 Basics . 11 2.2 Graphs . 11 2.3 Graph drawing . 13 2.4 Graph coloring . 14 2.4.1 List coloring . 15 2.4.2 Packing coloring . 15 3 Graph coloring techniques 16 3.1 The Four color theorem . 16 3.2 Degeneracy . 17 3.3 Kempe chains . 17 3.4 Identification . 19 3.5 Discharging . 20 3.6 Computer search . 22 3.7 Thomassen's technique . 23 4 6-critical graphs 25 4.1 Introduction . 25 4.2 6-critical graphs on the Klein bottle . 28 4.2.1 Introduction . 28 4.2.2 6-critical graphs . 29 4.2.3 Minimal graphs . 30 4.2.4 Embeddings of K6 on the Klein bottle . 34 4.2.5 Expansions of 2-cell embeddings of K6 ........... 35 4.2.6 Expansions of non-2-cell embedding of K6 ......... 37 6 4.2.7 Wrapping together . 40 4.3 6-critical graphs with few crossings . 41 4.3.1 Introduction . 41 4.3.2 Drawings with crossings . 41 4.3.3 Properties of 6-critical graphs . 43 4.3.4 6-critical graph of crossing number 5 . 45 4.3.5 Colouring graphs whose crossings are covered by few edges 46 4.3.6 5-colouring graphs with 4 crossings . 52 5 3-choosability of planar triangle-free graphs 60 5.1 Introduction . 60 5.2 Graphs without 3-,7-,8-cycles . 62 5.2.1 Reducible configurations . 62 5.2.2 Initial charges . 64 5.2.3 Discharging rules . 65 5.2.4 Final charges . 66 5.3 Graphs with constraints on 4-cycles . 69 5.3.1 Introduction . 69 5.3.2 Short outer face corollary . 69 5.3.3 Proof of Theorem 5.9 . 70 5.3.4 Vertices in the outer face . 75 6 Packing coloring 81 6.1 Introduction . 81 6.2 Density . 82 6.3 Square lattices . 84 6.3.1 Two layers are not colorable . 84 6.3.2 Lower bound 10 . 86 6.3.3 Lower bound 12 (computer assisted) . 90 6.4 Hexagonal lattice . 91 6.4.1 Upper bound 7 . 91 6.4.2 Coloring two layers . 91 6.4.3 Six layers are not colorable . 92 7 Conclusion 97 7 Bibliography 99 List of Figures 103 8 Chapter 1 Introduction Graph coloring is a popular topic of discrete mathematics. It has roots in the Four color problem which was the central problem of graph coloring in the last century. The Four color problem asks if it is possible to color every planar map by four colors. Although it is claimed to the Four color theorem has its roots in practice there is no evidence that the question was raised by map-makers [50]. It seems that mathematicians just asked themselves. Despite the theoretical origin the graph coloring has found many applications in practice like scheduling, frequency assignment problems, segmentation etc. A popular application in recent years is Sudoku. Many variants and generalizations of the graph coloring have been proposed since the Four color theorem. In this thesis, we present new results on graph coloring, list coloring and packing coloring. The next chapter contains a quick review of definitions used in this thesis. It is not a completely exhaustive list but it is covering the essential ones. Chap- ter 3 describes basic graph coloring methods which are demonstrated on simple theorems. It should help readers to follow proofs in the subsequent chapters. The first chapter with new results is dedicated to ordinary coloring and critical graphs. We present theorems enumerating all 6-critical graphs on the Klein bottle and all 6-critical graphs with crossing number at most four. The chapter is based on papers 6-critical graphs on the Klein bottle by Kawarabayashi, Kr´al',Kynˇcl and Lidick´y[31] (published in SIAM Journal on Discrete Mathematics) and 5- colouring graphs with 4 crossings by Erman, Havet, Lidick´yand Pangr´ac[17] (published in SIAM Journal on Discrete Mathematics). In Chapter 5 we study list coloring which is a generalization of coloring where every vertex has its own list of colors. We focus on 3-choosability of planar graphs and give two different conditions implying that a planar graph is 3-choosable. The chapter is based on papers On 3-choosability of plane graphs without 3-, 7- and 8-cycles [11] (published in Discrete Mathematics) and 3-choosability of triangle- free planar graphs with constraints on 4-cycles [12] (published in SIAM Journal on Discrete Mathematics) by Dvoˇr´ak,Lidick´yand Skrekovski.ˇ The last chapter deals with a recent concept called packing coloring. The concept is inspired by frequency assignment where some frequencies must be used more 9 sparsely than the others. We improve bounds for the square grid and hexagonal grid. The chapter is based on a paper The packing chromatic number of infinite product graphs by Fiala, Klavˇzarand Lidick´y[19] (published in European Journal of Combinatorics) and a manuscript The packing chromatic number of the square lattice is at least 12 Ekstein, Fiala, Holub and Lidick´y[13]. 10 Chapter 2 Definitions In this chapter we review definitions used in the following chapters. For more examples and details see books by Matouˇsekand Neˇsetˇril[35], Bollob´as[4] or Diestel [8]. 2.1 Basics We denote the set of natural numbers by N, the set of integers by Z, the set of rational numbers by Q and the set of reals by R. We abbreviate the set 1; 2; : : : n by [n]. For a set S we denote the set of all subsets of size k by S f g k 2.2 Graphs Graphs are the main objects studied in this thesis.