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Variations on a Graph Coloring Theme Western Michigan University ScholarWorks at WMU Dissertations Graduate College 6-2012 Variations on a Graph Coloring Theme Bryan Phinezy Western Michigan University Follow this and additional works at: https://scholarworks.wmich.edu/dissertations Part of the Mathematics Commons Recommended Citation Phinezy, Bryan, "Variations on a Graph Coloring Theme" (2012). Dissertations. 3400. https://scholarworks.wmich.edu/dissertations/3400 This Dissertation-Open Access is brought to you for free and open access by the Graduate College at ScholarWorks at WMU. It has been accepted for inclusion in Dissertations by an authorized administrator of ScholarWorks at WMU. For more information, please contact [email protected]. Variations on a Graph Coloring Theme by Bryan Phinezy A Dissertation Submitted to the Faculty of The Graduate College in partial fulfllment of the requirements for the Degree of Doctor of Philosophy Department of Mathematics Advisor: Ping Zhang, Ph.D. Western Michigan University Kalamazoo, Michigan June 2012 © 2012 Bryan Phinezy ACKNOWLEDGEMENTS I would like to thank my doctoral committee consisting of Dr. Ping Zhang, Dr. Gary Chartrand, Dr. Garry Johns, Dr. Allen Schwenk, and Dr. Arthur White. Without them, none of this would have happened. I would specificallylike to thank my committee chair, Dr. Ping Zhang for all of the time she spent with me to further my research endeavors. I would like to thank the staff in the math office for all of the help they have given me over the years. My classmates have been a big help for me to get through the program as well. Specifically, the people I share an office with have been around to listen to me as I worked on these problems. I would also like to thank my family for all of the support they have given me during this process. Bryan Phinezy 11 Variations on a Graph Coloring Theme Bryan Phinezy, Ph.D. Western Michigan University, 2012 Historically, the subject of graph colorings has been the most popular research area in graph theory. There are many problems in mathematics and in real life that can be represented by a graph and whose solution involves finding a specific coloring of this graph. Our research consists of two parts: (1) combinatorial problems and vertex colorings and (2) distance-defined colorings. In this research, we show that certain combinatorial puzzles and problems can be placed in a graph coloring setting and graph colorings can be defined in terms of distance in graphs that are useful in applications. Two vertices u and v in a nontrivial connected graph G are twins if u and v have the same neighbors in V (G) - { u,v}. If u and v are adjacent, they are referred to as true twins; while if u and v are nonadjacent, they are false twins. For a positive integer k, V(G) let c: -+ Zk be a vertex coloring where adjacent vertices may be assigned the V(G) same color. The coloring c induces another vertex coloring c' : -+ Zk defined by V(G), c'(v) = I::uEN[v] c(u) for each v E where N[v] is the closed neighborhood of v. Then c is called a closed modular k-coloring if c'( u) #- c'( v) in Zk for all pairs u, v of adjacent vertices that are not true twins. The minimum positive integer k for which G has a closed modular k-coloring is the closed modular chromatic number mc(G) of G. These concepts were inspired by two combinatorial problems. For an ordered set W = { w1,w 2, ... ,w k} of k distinct vertices in a nontrivial con­ nected graph G, the metric code of a vertex v of G with respect to W is the k-vector code(v) = (d(v,w1),d(v,w2), .. · ,d(v,wk)) where d( v, wi) is the distance between v and Wi for 1 � i � k. The set W is a local metric set of G if code( u) #- code( v) for every pair u, v of adjacent vertices of G. The minimum positive integer k for which G has a local metric k-set is the local metric dimension lmd(G) of G. A local metric set of G of cardinality lmd( G) is a local metric basis of G. These concepts were inspired by the well-studied concepts of metric sets and metric dimension in graphs and their applications. TABLE OF CONTENTS ACKNOWLEDGEMENTS .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ii LIST OF TABLES. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. V LIST OF FIGURES... .. .. .. .. .. .. .. .. .. .. .. .. .. .. VI CHAPTER 1. Introduction ........................................................ 1 1.1. History and Motivation . .. .. .. .. .. .. .. .. .. .. .. .. .. 1 1.2. A Checkerboard Problem ......................................... 3 1.3. The Lights Out Problem.......................................... 4 1.4. Distance-Defned Neighbor-Distinguishing Sets.................... 6 2. Preliminary Results on Closed Modular Colorings .............. 9 2.1. Terminology and Notation........................................ 9 2.2. Paths, Cycles and Complete Bipartite Graphs.................... 14 2.3. Regular Complete Multipartite Graphs........................... 20 2.4. Closed k-Modular Graphs.... .. .. .. .. .. .. .. .. .. .. .. .. 29 3. Closed Modular Colorings in 'ees .............................. 34 3.1. Preliminary Results. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 34 3.2. A Four Color Theorem.. .. .. .. .. .. .. .. .. .. .. 41 3.3. Rooted Tees . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 49 3.4. Odd Rooted Tees .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 51 iii Table of Contents - Continued CHAPTER 4. Closed Modular Colorings and Graph Operations .............. 62 4.1. Cartesian Products . .. .. .. .. .. ... .. .. .. .. .. .. .. .. 62 4.2. Coronas of Graphs. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 75 4.3. Joins of Graphs..... .. .. .. .. .. .. .. .. .. .. .. .. .. 78 5. Local Metric Dimension........................................... 90 5.1. Terminology and Notation....................................... 90 5.2. Graphs with Prescribed Order and Local Metric Dimension...... 92 5.3. Bounds for Local Metric Dimension.............................. 98 5.4. Uniqueness and Non-Uniqueness of Local Metric Bases.......... 104 5.5. Joins and Compositions of Graphs.............................. 107 5.6. Vertex and Edge Deletions...................................... 118 6. Upper Local Metric Dimension ................................. 124 6.1. Introduction .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 124 6.2. Comparing the Two Local Metric Dimensions................... 126 6.3. Realization Results .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 131 6.4. Local Metric Dimensions of Graphs with Prescribed Order . .. 135 7. Topics for Further Study ........................................ 140 BIBLIOGRAPHY.......................................................... 143 iv LIST OF TABLES 6.1 Values of lmd(G) and lmd+ (G) of all connected graphs of order 6.......... 136 V LIST OF FIGURES 1.1 A coin placement on a 3 x 6 checkerboard.................................... 4 1.2 Lights Out Game............................................................. 5 2.1 A graph G with x(G) = 2 and mc(G) = 3 ..........·......................... 10 2.2 A closed modular 3-coloring of the Petersen graph........................... 12 3.1 A tree T with mc(G)=3..................................................... 35 3.2 A bipartite graph with me(G) = 2.. .. .. .. .. .. .. .. .. .. .. .. 37 3.3 The tree Tk for k = 2........................................................ 47 3.4 A tree T with mc(T) = 3.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 55 3.5 A tree T* of type S = {2, 5} with mc(T*) = 2 . .. .. .. .. .. .. 56 3.6 A tree T with me( G) = 3.................................................... 57 4.1 An illustration of the coloring c in case 1.................................... 64 4.2 An illustration of the coloring c in case 2.................................... 64 4.3 An illustration of the coloring c in case 3 . .. .. .. .. .. .. .. .. .. .. .. .. .. 65 4.4 An illustration of the coloring c in case 4.................................... 66 4.5 A closed modular 3-coloring of CnDK2, for n = 5, 9.......................... 67 4.6 A closed modular 3-coloring of CnDK2, for n = 7, 11 . .. .. .. .. .. .. .. .. 68 4.7 A closed modular 2-coloring of CsDK2...................................... 68 4.8 A closed modular 3-coloring of C4□K2 and C6□K2 . .. .. .. .. .. .. 69 4.9 A closed modular 4-coloring of Cn + K1 for n = 5, 9......................... 87 + 4.10 A closed modular 4-coloring of C1 3 K1 ................................... 88 vi List of Figures - Continued 4. 11 A closed modular 4-coloring of Cn + K1 for n = 7, 11....................... 89 5.1 Metric sets.................................................................. 91 5.2 A graph with metric dimension 3 and local metric dimension 2 .. .. .. .. .. 92 6.1 A graph G with lmd(G) = 3 and dim(G) = lmd+ (G) = 4................... 125 6.2 Graphs F and H with lmd(F) < dim(F) < lmd+ (F) and lmd(H) < lmd+ (H) < dim(H)............................................. 126 6.3 A graph G with lmd(G) = 40 and lmd+ (G) = 224.......................... 134 6.4 Graphs G1 , Fi and H1 of order 6........................................... 135 vu Chapter 1 Introduction 1.1 History and Motivation In 1986, at the 250th Anniversary of Graph Theory Conference held at Indiana University­ Purdue University Fort Wayne, a weighting (or edge labeling with positive integers) of a connected graph G was introduced fr the purpose of producing a weighted graph whose degrees ( obtained by adding the weights of the incident edges of each vertex) were distinct. Such a weighted graph was called irregular. This concept could be looked
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