Iowa State University Capstones, Theses and Graduate Theses and Dissertations Dissertations 2021 Turan problems in extremal graph theory and flexibility Kyle Edinger Murphy Iowa State University Follow this and additional works at: https://lib.dr.iastate.edu/etd Recommended Citation Murphy, Kyle Edinger, "Turan problems in extremal graph theory and flexibility" (2021). Graduate Theses and Dissertations. 18567. https://lib.dr.iastate.edu/etd/18567 This Dissertation is brought to you for free and open access by the Iowa State University Capstones, Theses and Dissertations at Iowa State University Digital Repository. It has been accepted for inclusion in Graduate Theses and Dissertations by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. Tur´anproblems in extremal graph theory and graph flexibility by Kyle Edinger Murphy A dissertation submitted to the graduate faculty in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Major: Mathematics Program of Study Committee: Bernard Lidick´y,Major Professor Steve Butler Claus Kadelka Ryan Martin Michael Young The student author, whose presentation of the scholarship herein was approved by the program of study committee, is solely responsible for the content of this dissertation. The Graduate College will ensure this dissertation is globally accessible and will not permit alterations after a degree is conferred. Iowa State University Ames, Iowa 2021 Copyright © Kyle Edinger Murphy, 2021. All rights reserved. ii DEDICATION I would like to dedicate this thesis to my family Jim, Gail and Sara, and my fianc´eeVirginia. The support from each of them provided me throughout graduate school. iii TABLE OF CONTENTS Page ACKNOWLEDGMENTS . .v ABSTRACT . vi CHAPTER 1. GENERAL INTRODUCTION . .1 1.1 Extremal Graph Theory . .2 1.2 Graph Coloring . .4 1.2.1 Flexibility . .5 1.2.2 The Discharging Method . .6 1.3 References . .7 CHAPTER 2. MAXIMIZING FIVE-CYCLES IN Kr-FREE GRAPHS . .9 2.1 Abstract . .9 2.2 Introduction . .9 2.2.1 The Flag Algebra Method . 14 2.3 Proof of Theorem 2.2.4(i) ................................. 17 2.3.1 Finding the Optimal Bound . 23 2.4 Stability . 24 2.5 Exact Result . 31 2.6 Conclusion . 39 2.7 References . 40 2.8 Appendix . 43 2.8.1 Proof of Claim 2.3.1................................ 44 2.8.2 SageMath code for Claim 2.4.10......................... 52 2.8.3 SageMath code for Claim 4.5 . 54 2.8.4 SageMath code for Claim 4.7 . 56 CHAPTER 3. PATHS OF LENGTH THREE ARE Kr+1-TURAN´ GOOD . 58 3.1 Abstract . 58 3.2 Introduction . 58 3.2.1 Background and Conventions . 61 3.2.2 The Flag Algebra Method . 62 3.3 Theorem 3.2.3(i)...................................... 65 3.4 Stability . 71 3.5 Exact Result . 79 3.6 Concluding Remarks . 88 3.7 Acknowledgements . 89 3.8 References . 89 iv 3.9 Appendix . 91 CHAPTER 4. ON WEAK FLEXIBILITY OF PLANAR GRAPHS . 95 4.1 Abstract . 95 4.2 Introduction . 96 4.3 Methods - informal discussion . 99 4.4 Methods - definitions and lemmas . 101 4.5 Proof of Theorem 4.2.1................................... 108 4.5.1 Reducible Configurations . 108 4.5.2 Discharging . 113 4.6 Proof of Theorem 4.2.2................................... 122 4.6.1 Reducible configurations . 122 4.6.2 Discharging rules . 125 4.7 References . 134 CHAPTER 5. GENERAL CONCLUSIONS . 136 v ACKNOWLEDGMENTS I want to thank everyone who helped contribute to my graduate work at Iowa State. Thank you to Bernard Lidick´yfor his guidance and friendship throughout my time here. I would also like to thank my committee members for their time and effort. Thank you to Tom´aˇsMasaˇr´ık,J.D. Nir, and Shira Zerbib for your collaborations and contributions. An additional thank you to Dr. Teresa Haynes and Dr. Robert Beeler for your guidance early in my career, and for inspiring me to get my Ph.D. Finally, I would like to especially thank my mom, who always encouraged and supported my interest in math. Without her help I would not have been able to do this. vi ABSTRACT In this work we will study two distinct areas of graph theory: generalized Tur´anproblems and graph flexibility. In the first chapter, we will provide some basic definitions and motivation. Chapters 2 and 3 contain two submitted papers showing that two graphs, the cycle on five vertices and the path on four vertices, are maximized by the Tur´angraph when forbidding sufficiently large cliques. In chapter 4, we present new results on graph flexibility and weak flexibility for planar graphs. 1 CHAPTER 1. GENERAL INTRODUCTION This work focuses on two distinct areas of graph theory: extremal graph theory and graph coloring. In Chapters 2 and 3 we will focus on extremal graph theory, using the flag algebra method in order to calculate the generalized Tur´annumbers of two different graphs. In Chapter 4, we will turn our attention to graph coloring, presenting new results in the area of flexibility, which is a variant of list coloring. In this chapter, we will introduce some necessary definitions, along with background on each of the two subjects. Each of the chapters in this thesis contains a different paper which has been submitted for publication. A graph G is a pair G = (V (G);E(G)) where V (G) is a set whose elements are called vertices, and E(G) is a set containing pairs of elements in V (G), or edges. Two vertices connected by an edge are said to be adjacent. The degree of a vertex v, often denoted d(v), is the number of vertices in V (G) that are adjacent to v. The following well-known theorem is commonly referred to as the \Handshaking Lemma." X Theorem 1.0.1. If G is a graph with m edges, then d(v) = 2m. v2V (G) It is common to depict a graph using points drawn in the plane to represent vertices, and lines drawn between those points to represent edges. v2 V (G) = fv1; v2; v3; v4; v5g v3 v1 E(G) = fv1v2; v2v3; v3v4; v4v5; v5v1g v4 v5 Figure 1.1 A graph G = (V (G);E(G)) A graph is said to be simple if it does not contain any multiedges or loops. A multiedge is a pair of vertices connected by more than one distinct edge, and a loop is an edge of the form vv for 2 some vertex v. Moving forward, we will assume that all graphs being discussed are simple. In a graph G, two vertices u and v are called connected if there exists a sequence of edges of the following form vx1; x1x2; : : : ; xi−1xi; xiu. In other words, there exists a path from v to u. A graph G is said to be connected if every pair of vertices in G are connected. If G is not connected, then we say that it is disconnected. Unless specified otherwise, all graphs in this manuscript are connected. Let G be a graph and let C be a set of colors. A proper coloring of G is a map φ : V (G) ! C such that for each edge uv 2 E(G), φ(u) 6= φ(v). If jCj = k and such a map exists, then we say that G is k-colorable. The smallest value of k for which G is k-colorable is denoted χ(G) and called the chromatic number of G. Note that χ(G) exists for each graph G, since one can always let jCj = jV (G)j and simply let φ be any bijective map from V (G) to C. Two graphs G = (V (G);E(G)) and H = (V (G);E(G)) are said to be isomorphic if there exists a bijection f : V (G) ! V (H) such that two vertices u; v 2 V (G) are adjacent if and only if f(u) and f(v) are adjacent in V (H). A subgraph F = (V (F );E(F )) of a graph G = (V (G);E(G)) is a graph such that V (F ) ⊆ V (G) and E(F ) ⊆ E(G) so that for each edge uv 2 E(F ), both u and v are contained in V (F ). A subgraph F is said to be induced if E(F ) contains all of the edges in E(G) for which both vertices are contained in V (F ). 1.1 Extremal Graph Theory Extremal graph theory focuses on maximizing or minimizing some function on a graph subject to some constraint. It has applications in computer science, and is useful for studying the properties of large graphs like the internet. In this thesis we will study generalized Tur´an problems, which deal with maximizing the frequency of a particular subgraph H in some host graph G while forbidding any occurrence of a different subgraph F . The forbidden graph F is most commonly a complete graph Kr, which is the graph on r vertices in which each pair of vertices is adjacent. The first result in extremal graph theory is Mantel's Theorem [7], concerning the graph K3. 3 Theorem 1.1.1 (Mantel's Theorem). The maximum number of edges among all K3-free graphs n2 on n vertices is . 4 Later, Tur´an[8] generalized this result to all Kr-free graphs for r ≥ 4. In a moment we will state Tur´an'sTheorem, but first we will require some definitions. A graph G is said to be r-partite if V (G) can be partitioned into r sets V1;V2;:::;Vr so that every edge of G joins two vertices in distinct sets.