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Chromatic Polynomials for Graphs with Split Vertices CHROMATIC POLYNOMIALS FOR GRAPHS WITH SPLIT VERTICES Sarah E Adams A Thesis Submitted to the Graduate College of Bowling Green State University in partial fulfillment of the requirements for the degree of MASTERS OF ART August 2020 Committee: Mihai Staic, Advisor Juan Bes Kim Rogers Copyright c August 2020 Sarah E Adams All rights reserved iii ABSTRACT Mihai Staic, Advisor Graph theory is a branch of mathematics that uses graphs as a mathematical structure to model relations between objects. Graphs can be categorized in a wide variety of graph families. One im- portant instrument to classify graphs is the chromatic polynomial. This was introduced by Birkhoff in 1912 and allowed to further study and develop several graph related problems. In this thesis, we study some problems that can be approached using the chromatic polynomial. In the first chapter, we introduce general definitions and examples of graphs. In the second chapter, we talk about graph colorings, the greedy algorithm, and give a short description for the four color problem. In the third chapter, we introduce the chromatic polynomial, study its property, and give some examples of computations. All of these are classical results. In chapter 4, we introduce colorings of graphs with split vertices, and give an application to the scheduling problem. Also, we show how the chromatic polynomial can be used in that setting. This is our “semi-original” contribution. Finally, in the last chapter, we talk about distance two colorings for graphs, and give examples on how this applies to coloring maps. iv ACKNOWLEDGMENTS Foremost, I would like to thank my committee members, Dr. Kim Rogers and Dr. Juan Bes, for taking the time to give their insights and guidance throughout my years in graduate school. Ad-ditionally, I would like to express my sincere gratitude to Dr. Mihai Staic for being an excellent advisor. He was not only an excellent advisor, but an excellent teacher and role model. He provided me with support and encouragement that made me want to learn more about mathematics. Addi-tionally, I am grateful for my colleagues for their continued encouragement and enthusiasm over the last two years. Finally, I can’t thank my my friends and family enough for their unconditional love. v TABLE OF CONTENTS Page CHAPTER 1 GRAPHS . 1 1.1 Defining Graphs . 1 1.2 Common Families of Graphs . 4 CHAPTER 2 GRAPH COLORING . 7 2.1 Definitions and Examples . 8 2.2 Coloring Algorithms . 11 2.3 Greedy Algorithm Examples . 12 2.4 The Four Color Theorem . 16 CHAPTER 3 CHROMATIC POLYNOMIALS . 19 3.1 History of the Chromatic Polynomial . 19 3.2 Defining Chromatic Polynomials . 19 3.3 Chromatic Polynomials of Simple Graphs . 20 3.4 Edge Deletion and Contraction . 22 CHAPTER 4 SPLIT VERTICES . 27 4.1 Scheduling . 27 4.2 Definitions . 29 4.3 The Split Chromatic Polynomial . 30 CHAPTER 5 NEIGHBORING VERTICES . 39 5.1 Introduction . 39 5.2 Definitions . 41 5.3 Examples . 42 CHAPTER 6 CONCLUSIONS . 46 6.1 Main Results . 46 vi 6.2 Related Questions . 46 BIBLIOGRAPHY . 48 vii LIST OF FIGURES Figure Page 1.1 A graph with 6 vertices . 2 1.2 An example of a graph with multi-edges . 3 1.3 An example of a graph with a self-loop . 3 1.4 An example of a simple graph . 3 1.5 An example of an empty graph with 3 nodes . 4 1.6 The graphs K3 and K4 ................................ 4 1.7 A bipartite graph . 5 1.8 The P4 graph . 5 1.9 The S6 graph . 6 1.10 The C4 graph . 6 2.1 South America Mapping . 7 2.2 The graph of South America . 8 2.3 Proper Coloring of P5 graph . 9 2.4 Proper Coloring of K5 graph . 9 2.5 Proper Coloring of an empty graph . 10 2.6 Proper Coloring of an C5 graph . 11 2.7 Greedy Coloring of K3 graph . 12 2.8 Greedy Coloring of Graph G using three colors . 13 2.9 Greedy Coloring of Graph G using four colors . 14 2.10 Another example of greedy coloring . 15 2.11 Another example of greedy coloring . 16 2.12 The S5 graph . 16 2.13 United States Map using Four Colors . 17 3.1 E3 graph . 20 viii 3.2 Coloring of P4 graph . 21 3.3 Proper coloring of K5 graph . 22 3.4 An example of edge deletion, e = (1; 3) ....................... 23 3.5 Contraction of the edge (1,4) . 23 3.6 The graph C4 ..................................... 25 3.7 C4 − e = P4 and C4=e = K3 ............................. 25 3.8 A graph G, e = (2; 4) ................................. 26 3.9 G − e = C4 and G=e = P3 .............................. 26 4.1 Committees Graph Coloring . 28 4.2 Committees with Split Vertices . 30 4.3 Split Coloring . 31 4.4 DP3 .......................................... 32 4.5 G = DP3 ....................................... 33 4.6 The graph of G after one deletion and contraction . 33 4.7 The 3 types of graphs of G with edge contraction and deletion . 33 4.8 Split Coloring of E3 .................................. 34 4.9 DE3 .......................................... 34 4.10 Split Coloring for K5 ................................. 34 4.11 DK5 = K10 ...................................... 35 4.12 Split Coloring of S4 .................................. 36 4.13 Split Coloring of S5 .................................. 37 4.14 DS4 .......................................... 37 4.15 DS5 .......................................... 38 5.1 Map of Midwest . 39 5.2 Graph of Midwest . 40 5.3 Graph of Midwest with New Condition . 41 ix 5.4 Distance Two Graph Coloring of Midwest . 41 5.5 The S6 graph . 43 5.6 NS6 = K6 graph . 43 5.7 Graph Coloring with distance 2 coloring . 44 5.8 Graph G and graph NG ................................ 44 x LIST OF TABLES Table Page 4.1 Schedule 1 . 27 4.2 Schedule 2 . 29 4.3 Summary of Chromatic Polynomials of Graphs . 38 1 CHAPTER 1 GRAPHS There is a vast array of configurations that one can make by connecting vertices. These con- nections are seen in everyday life in roadways between cities, scheduling, networks, land lines, electric grids, and more. Additionally, these connections with vertices can be used to represent re- lationships within society and ecosystems. These are just a few ways in which we can use graphs. 1.1 Defining Graphs In this section, we will give some general definitions and examples of graphs. We will start by recalling a few definitions and examples from Beineke and Wilson (2015) and from Gross, Yellen, and Zhang (2014). Definition 1.1.1. A graph G is a pair of sets (V; E), where V is a finite non-empty set of ele- ments called vertices, and E is a finite set of elements called edges. Moreover, each edge has two associated vertices. The sets V and E are the vertex-set and edge-set of G, and are sometimes denoted by V (G) and E(G). The number of vertices in G is called the order of G and is usually denoted by n(G). The number of edges is denoted by m(G). Definition 1.1.2. a) If a vertex v is an endpoint of the edge e, then v is said to be incident on e, and e is incident on v. b) A vertex u is adjacent to vertex v if they are joined by an edge. Two adjacent vertices are called neighbors. c) Adjacent edges are two edges that have an endpoint in common. 2 Example 1.1.3. Referring to Figure 1.1, the graph has vertices V = f1; 2; 3; 4; 5; 6g and edges E = fa = (1; 2); b = (1; 5); c = (2; 3); d = (2; 5); e = (3; 4); f = (4; 5); g = (4; 6)g. The vertex 3 is adjacent to both vertices 2 and 4. Additionally, we can say 3 and 2 are neighbors. An example of an adjacent edge is the edge c = (2; 3) that connects vertices 2 and 3, and the edge e = (3; 4) that connects by vertices 3 and 4. Notice that the common endpoint for these edges is 3. 6 5 4 1 2 3 Figure 1.1 A graph with 6 vertices Definition 1.1.4. a) A proper edge is an edge that joins two distinct vertices. b) A multi-edge is a collection of two or more edges having the same endpoints. c) A simple adjacency between vertices occurs when there is exactly one edge between them. d) The edge-multiplicity between a pair of vertices u and v is the number of edges between them. e) A self-loop is an edge that joins a single vertex to itself. f) A simple graph is a graph that has no self-loops or multi-edges. 3 1 2 3 4 Figure 1.2 An example of a graph with multi-edges 4 1 2 3 loop Figure 1.3 An example of a graph with a self-loop 4 1 2 3 Figure 1.4 An example of a simple graph Example 1.1.5. 1. We can see in Figure 1.2 that there are two edges having the same endpoints. For example there are two edges connecting 2 and 4. We call this a multi-edge. Since there are two edges, we say that we have edge-multiplicity 2. 2. In Figure 1.3 we have a graph with a self-loop on vertex 3. 3. In Figure 1.4 we have an example of a simple graph (i.e. no loops and no multi-edges). Convention 1.1.6. In this paper all graphs have the property that between two distinct vertices we have at most one edge.
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