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A Study on Graph Coloring and Digraph Connectivity View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by The Research Repository @ WVU (West Virginia University) Graduate Theses, Dissertations, and Problem Reports 2018 A Study on Graph Coloring and Digraph Connectivity Murong Xu Follow this and additional works at: https://researchrepository.wvu.edu/etd Recommended Citation Xu, Murong, "A Study on Graph Coloring and Digraph Connectivity" (2018). Graduate Theses, Dissertations, and Problem Reports. 6990. https://researchrepository.wvu.edu/etd/6990 This Dissertation is protected by copyright and/or related rights. It has been brought to you by the The Research Repository @ WVU with permission from the rights-holder(s). You are free to use this Dissertation in any way that is permitted by the copyright and related rights legislation that applies to your use. 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A Study on Graph Coloring and Digraph Connectivity Murong Xu Dissertation submitted to the Eberly College of Arts and Sciences at West Virginia University in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics Hong-Jian Lai, Ph.D., Chair John Goldwasser, Ph.D. Guodong Guo, Ph.D. Rong Luo, Ph.D. Jerzy Wojciechowski, Ph.D. Department of Mathematics Morgantown, West Virginia 2018 Keywords: r-hued coloring, linear list r-hued coloring, edge-connectivity, restricted edge-connectivity, strong arc connectivity Copyright 2018 Murong Xu ABSTRACT A Study on Graph Coloring and Digraph Connectivity Murong Xu This dissertation focuses on coloring problems in graphs and connectivity problems in di- graphs. We obtain the following advances in both directions. 1. Results in graph coloring. For integers k; r > 0, a (k; r)-coloring of a graph G is a proper coloring on the vertices of G with k colors such that every vertex v of degree d(v) is adjacent to vertices with at least minfd(v); rg different colors. The r-hued chromatic number, denoted by χr(G), is the smallest integer k for which a graph G has a (k; r)-coloring. For a k-list assignment L to vertices of a graph G, a linear (L; r)-coloring of a graph G is a coloring c of the vertices of G such that for every vertex v of degree d(v), c(v) 2 L(v), the number of colors used by the neighbors of v is at least minfdG(v); rg, and such that for any two distinct colors i and j, every component of G[c−1(fi; jg)] must be a path. The linear list r-hued ` chromatic number of a graph G, denoted χL;r(G), is the smallest integer k such that for every k-list L, G has a linear (L; r)-coloring. Let Mad(G) denotes the maximum subgraph average degree of a graph G. We prove the following. (i) If G is a K3;3-minor free graph, then χ2(G) ≤ 5 and χ3(G) ≤ 10. Moreover, the bound of χ2(G) ≤ 5 is best possible. (ii) If G is a P4-free graph, then χr(G) ≤ χ(G) + 2(r − 1), and this bound is best possible. (iii) If G is a P5-free bipartite graph, then χr(G) ≤ rχ(G), and this bound is best possible. (iv) If G is a P5-free graph, then χ2(G) ≤ 2χ(G), and this bound is best possible. (v) If G is a graph with maximum degree ∆, then each of the following holds. 7 ` (i) If ∆ ≥ 9 and Mad(G) < 3 , then χL;r(G) ≤ maxfd∆=2e + 1; r + 1g. 12 ` (ii) If ∆ ≥ 7 and Mad(G) < 5 , then χL;r(G) ≤ maxfd∆=2e + 2; r + 2g. 5 ` (iii) If ∆ ≥ 7 and Mad(G) < 2 , then χL;r(G) ≤ maxfd∆=2e + 3; r + 3g. ` (vi) If G is a K4-minor free graph, then χL;r(G) ≤ maxfr; d∆=2eg + d∆=2e + 2. ` (vii) Every planar graph G with maximum degree ∆ has χL;2(G) ≤ ∆ + 7. 2. Results in digraph connectivity. For a graph G, let κ(G), κ0(G), δ(G) and τ(G) denote the connectivity, the edge-connectivity, the minimum degree and the number of edge-disjoint spanning trees of G, respectively. Let f(G) denote κ(G), κ0(G), or δ(G), and define f(G) = maxff(H): H is a subgraph of Gg. An edge cut X of a graph G is restricted if X does not contain all edges incident with a vertex in G. The restricted edge-connectivity of G, denoted by λ2(G), is the minimum size of a restricted edge-cut of G. We define λ2(G) = maxfλ2(H): H ⊆ Gg. For a digraph D, let κ(D), λ(D), δ−(D), and δ+(D) denote the strong connectivity, arc-strong connectivity, minimum in-degree, and out-degree of D, respectively. For each f 2 fκ, λ, δ−; δ+g, define f(D) = maxff(H): H is a subdigraph of Dg. Catlin et al. in [Discrete Math., 309 (2009), 1033-1040] proved a characterization of κ0(G) in terms of τ(G). We proved a digraph version of this characterization by showing that a digraph D is k-arc-strong if and only if for any vertex v in D, D has k-arc-disjoint spanning arborescences rooted at v. We also prove a characterization of uniformly dense digraphs analogous to the characterization of uniformly dense undirected graphs in [Discrete Applied Math., 40 (1992) 285 - 302]. For an integer k > 0, a simple digraph D with jV (D)j ≥ k + 1 is k-maximal if every subdigraph H of D satisfies λ(H) ≤ k, but adding a new arc to D results in a subdigraph H0 with λ(H0) ≥ k +1. We prove that if D is a simple k-maximal digraph on n > k +1 ≥ 2 vertices, then n n k + 2 jA(D)j ≥ + (n − 1)k + b c 1 + 2k − : 2 k + 2 2 This bound is best possible. Furthermore, all extremal digraphs reaching this lower bound are characterized. Matula in [SIAM J. Appl. Math. 22 (1972) 459-480] initiated the study of f(G). He proved two minimax theorems related to δ(G) and κ0(G). We obtain analogous minimax duality results for λ2(G). Acknowledgements First and foremost, I am most indebted to my supervisor, Dr. Hong-Jian Lai, for his con- tinued encouragement and support over these last few years. It is a pleasure to work under his supervision. Without him, this paper could not have come about. I would like to take this opportunity to thank Dr. Suohai Fan at Jinan University (Guangzhou, China) who brought me to the research field of combinatorics and graph theory. I would also like to thank my other committee members: Dr. John Goldwasser, Dr. Guodong Guo, Dr. Rong Luo, and Dr. Jerzy Wojciechowski, for their help during my studies. And finally, I would like to thank the Department of Mathematics and Eberly College of Arts and Sciences at West Virginia University for providing me with an excellent study environment and support during my study as a graduate student. iv DEDICATION To my mother Chongyu Fu, my father Xiaoping Xu and my husband Xin Luo v Contents 1 Introduction 1 1.1 Notations and Terminology . 1 1.2 Background on Graph Coloring . 2 1.3 Background on Digraph Connectivity . 5 2 On r-hued Coloring of Graphs without K3;3-minors 7 2.1 Main Results . 7 2.2 r-hued Chromatic Number of Graphs without K3;3-minors . 7 2.3 Dynamic Chromatic Number of Graphs without K4-minors . 11 3 On r-hued Coloring of Graphs without Short Induced Paths 14 3.1 Main Results . 14 3.2 On r-hued Colorings of P4-free Graphs . 14 3.3 On r-hued Colorings of P5-free Graphs . 18 3.3.1 On r-hued Colorings of P5-free Bipartite Graphs . 21 3.3.2 On 2-hued Colorings of P5-free Graphs . 24 3.4 Future Studies . 25 4 Linear List r-hued Colorings of Graphs with Bounded Maximum Subgraph Average Degrees 26 4.1 Main Results . 26 4.2 The Linear r-hued Chromatic Number of Certain Graph Families . 26 4.3 Linear 2-hued Colorings of Graphs with Bounded Average Degree . 30 5 Linear List r-hued Coloring of K4-minor Free Graphs 40 5.1 Main Results . 40 5.2 Proof of Theorem 5.1.1 . 40 5.3 Proof of Theorem 5.1.2 . 44 vi 6 Disjoint Spanning Arborescences in k-arc-strong Digraphs 46 6.1 Main Results . 46 6.2 Relationship Between k-arc-strong Connectivity and Disjoint Spanning Arbores- cences . 46 6.3 Characterization of Uniformly Dense Digraphs . 51 7 On the Lower Bound of k-maximal Digraphs 54 7.1 The Problem . 54 7.2 Properties of k-maximal Digraphs . 56 7.3 Structure of k-maximal Digraphs . 57 7.4 The Extremal Function . 67 8 Minimax Properties of Some Density Measures in Graphs and Digraphs 70 8.1 Introduction . 70 8.2 Slicing of Digraphs . 72 8.3 A Minimax Theorem in Restricted Edge-connectivity . 74 8.3.1 Restricted Slicing of a Graph . 75 8.3.2 A Minimax Theorem of Restricted Edge-connectivity . 77 8.4 Future Studies .
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