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E-Positivity Conjecture and Line Graphs by David E

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E-Positivity Conjecture and Line Graphs by David E E-POSITIVITY CONJECTURE AND LINE GRAPHS BY DAVID E NICHOLS A Thesis Submitted to the Graduate Faculty of WAKE FOREST UNIVERSITY GRADUATE SCHOOL OF ARTS AND SCIENCES in Partial Fulfillment of the Requirements for the Degree of MASTER OF ARTS Mathematics & Statistics May 2020 Winston-Salem, North Carolina Approved By: Sarah Mason, Ph.D., Advisor Edward Allen, Ph.D., Chair Hugh Howards, Ph.D. Acknowledgments I wish to express my deep and sincere gratitude to my research advisor, Dr. Sarah Mason, for her continual support, encouragement, and guidance through the entirety of the research process. I would like to thank my friends and classmates, Andres Guerrero-Guzman, De- siree Martin, and Heidi Whiteside, for their support both personally and academically. Finally, I would like to thank my parents, Fred and Tami Nichols, for their sacri- fices that allowed me to pursue an education. ii Table of Contents Acknowledgments . ii Abstract . iv Chapter 1 Introduction . 1 Chapter 2 Background . 3 Chapter 3 Line Graphs. 11 Chapter 4 Modified Line Graphs . 21 Chapter 5 Incomparability Graphs . 27 Chapter 6 Future Work . 38 Bibliography . 39 Appendix A Connected Simple Graphs with 6 Vertices. 40 Appendix B Edge Chromatic Polynomial Calculator . 44 Curriculum Vitae . 47 iii Abstract David Nichols Motivated by Richard Stanley's e-positivity conjecture, we explore the relationships between line graphs, modified line graphs, and incomparability graphs. We show that line graphs and modified line graphs will fail to contain the set of all incomparability graphs of (3+1)-free posets. In an effort to start finding a list of forbidden graphs for incomparability graphs, we test all simple, connected graphs with 6 or fewer vertices. iv Chapter 1: Introduction We will demonstrate that the use of line graphs and modified line graphs will not provide insight on Richard Stanley's e-positivity conjecture. As such, we will begin collecting graphs for a preliminary list of forbidden graphs of incomparability graphs. In Richard Stanley's \A Symmetric Function Generalization of the Chromatic Polynomial of a Graph" [6], he gave the following conjecture: Conjecture 1.0.1. If a poset, P , is (3 + 1)-free, then inc(P ) is e-positive. We refer to this conjecture as the e-positivity conjecture. This conjecture is sup- ported by Vesselin Gasharov's result that the incomparability graphs of (3+1)-free posets are all s-positive [2]. In an effort to gain insight into Stanley's e-positivity conjecture, we attempt to find collections of graphs which contain all incomparability graphs of (3+1)-free posets. Incomparability graphs of (3+1)-free posets are claw-free [6]. Thus, a natural collection of graphs to study is the collection of claw-free graphs. The graphs in this collection are not e-positive. For example, the 3-Sun graph as shown in Fig 1.1 is not e-positive. • • • • • • Figure 1.1: 3-Sun Graph 1 While claw-free graphs are not necessarily e-positive, they could still be considered a primary candidate for study in order to gain insight on Stanley's conjecture. For our purposes, we will first attempt to find a refinement of the collection of claw-free graphs. Through the work of Frank Harary, we can observe that line graphs are a subset of claw free graphs [3]. Line graphs would be an ideal candidate for study as Hassler Whitney proved that if the line graphs of two connected graphs are isomor- phic, then their underlying graphs will be isomorphic with only one exception; the triangle graph (K3) and the claw graph (K3;1) have isomorphic line graphs, but are not isomorphic [7]. We show that line graphs will fail to contain incomparability graphs of (3+1)-free posets. In an effort to recover the missing incomparability graphs, we modify the definition of line graphs. We refer to this new collection of graphs as modified line graphs. This new collection of graphs also fails to recover all of the incomparability graphs. While neither the collection of line graphs nor the collection of modified line graphs contain the collection of incomparability graphs, their study promotes the idea of attempting to find the forbidden graphs for incomparability graphs. In an effort to start building the list of forbidden graphs, we show that there are only five forbidden graphs with 6 or less vertices for modified line graphs (shown in Chapter 5). Furthermore, we prove that all graphs which contain Cn for n 5 will fail to be ≥ incomparability graphs. 2 Chapter 2: Background In order to understand Stanley's claim that all incomparability graphs of (3+1)- free posets are e-positivity, we first must first understand what a graph is. In gen- eral, a graph is simply a collection of vertices and a collection of edges which repre- sent some relationship between the vertices. We denote a graph as an ordered pair, G = (V (G);E(G)), where V (G) is the set of vertices and E(G) is the set of edges. We may use V and E respectively when the context is clear. The set of edges must satisfy the condition that an edge starts at a vertex and ends at a vertex with no vertices in-between. We say that two vertices are adjacent provided there is an edge connecting them. In the case that an edge starts and ends at the same vertex, we refer to the edge as a loop. In the case that two vertices have multiple edges between them, we refer to the collection of these edges as multi-edges. Furthermore, we refer to graphs that do not contain any loops or multi-edges as simple graphs. As an example, we consider the Collatz conjecture [4]. In the Collatz conjecture, we create a sequence starting at any natural number. The sequence will follow the pattern that if an is even, then an+1 = an=2, and if an is odd, then an+1 = 3an + 1. The conjecture states that any sequence that follows the described pattern will always reach 1. With this sequence in mind, we can let our set of vertices be the natural numbers. We will let there be an edge between two vertices, vi and vj, if and only if one of the following is true vi is even and vi = 2vj • vi is odd and 3vi + 1 = vj • With these requirements, we can create a graph that allows a visual representation 3 3 • 5 10 20 • • • 1 2 4 8 16 • • • • • 32 • Figure 2.1 of the Collatz conjecture as shown in Fig 2.1. In this example, the vertices are labeled with the natural number that they represent. We may wish to encode more information into the graph. Namely, it would be natural to want to show the flow of the sequence by giving a direction to each edge. Graphs in which we encode edges with directions are called directed graphs or di- graphs. Other common methods of adding information into graphs include assigning weights to each edge (called a weighted graph) or by assigning colors to either the vertices or the edges of a graph. In the case of the e-positivity conjecture, we will be focusing on vertex coloring. 1 5 2 • • • 1 Figure 2.2: Weighted Digraph 4 Definition 1. Let G be a graph. A (vertex) coloring is a function κ : V (G) C. It ! is common to let C be a set of colors or a subset of the natural numbers. where C is some set. We define a proper coloring of G as the same function with the caveat that for vi; vj V (G), κ(vi) = (vj) whenever vi is adjacent to vj. 2 6 To illustrate this definition, we will refer to Fig 2.3. In this example, we have a coloring since each vertex has been assigned a number, but it fails to be a proper coloring since there are two adjacent vertices which are both assigned the value 2. Changing the value of the central vertex to a 3 will make the coloring a proper coloring. 2 • 1 •2 1 • • Figure 2.3 A natural question for a given graph is \How many ways can we properly color its vertices?" The answer to this question motivates what we will call the chromatic polynomial of a graph. Definition 2. Let G be a graph and K be the set of all possible proper coloring functions κ : V (G) N. The chromatic polynomial of G, denoted XG(x), is given by ! X Y XG(x) = xκ(vi): κ2K vi2V (G) If we wish to restrict the number of colors such that κ : V (G) 1; 2; : : : ; n , we will ! f g describe the chromatic polynomial as XG(x1; x2; : : : ; xn): 5 • • • Figure 2.4: Complete graph; K3 The definition of chromatic polynomial does not impose an ordering on the ver- tices. As such, chromatic polynomials are symmetric. Consider the complete graph on three vertices shown in Fig 2.4. In this graph, all vertices are adjacent to each other. This implies that every proper coloring of this graph must use three distinct colors. Furthermore, observe that for any given proper coloring, permuting the colors of the vertices produces another proper coloring. This implies that X XK3 (x) = 6xixjxk: i<j<k2N It is important to note that non-isomorphic graphs may have the same chromatic polynomial as shown in Fig. 2.5. Therefore, we cannot construct a graph solely from its chromatic polynomial. • • • • • • • • • • • • Figure 2.5: Non-isomorphic graphs[5] with chromatic polynomial 720m1;1;1;1;1;1 + 216m2;1;1;1;1 + 60m2;2;1;1 + 122;2;2 + 24m3;1;1;1 + 6m3;2;1.
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