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. ∈ 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).

κ∈K vi∈V (G)

If we wish to restrict the number of colors such that κ : V (G) 1, 2, . . . , n , we will → { } 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

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.

The chromatic polynomial can be very cumbersome. To express this information in a more compact manner, we may wish to use a different basis to express the in-

6 formation. There are several bases which we can be used, but we will focus on two: monomial symmetric polynomials and elementary symmetric polynomials.

The monomial symmetric polynomials are a natural basis for chromatic polyno- mials as they are polynomials which contain all permutations of powers. Letting

λ = λ1λ2 . . . λl be a partition, we define the monomial symmetric polynomials as

X m (x) = xφ1 xφ2 . . . xφl λ λ1 λ2 λl φ∼λ where φ λ means that φ is a permutation of λ. As with the chromatic polynomial, ∼ we can restrict apply a restriction to the number of possible colors allowed. As an example,

4 3 4 3 3 4 3 4 4 3 3 4 m4,3,1(x1, x2, x3) = x1x2x3 + x1x2x3 + x1x2x3 + x1x2x3 + x1x2x3 + x1x2x3.

Another options is to use the elementary symmetric polynomials as a basis. Given a partition λ = λ1λ2 . . . λl, we define the elementary symmetric polynomials as

l Y eλ(x) = eλi (x) i=1 where X e (x) = x x . . . x . λi j1 j2 jλi

j1

To illustrate this definition, we will consider e2,1(x1, x2, x3). By part two of the defini- tion for elementary symmetric polynomials, we have e2(x1, x2, x3) = x1x2+x1x3+x2x3 and e1(x1, x2, x3) = x1 + x2 + x3. This gives

7 e2,1(x1, x2, x3) = e2(x1, x2, x3)e1(x1, x2, x3)

= (x1x2 + x1x3 + x2x3)(x1 + x2 + x3)

2 2 2 2 2 2 = x1x2 + x1x3 + x1x2 + x2x3 + x1x3 + x2x3 + 3x1x2x3

These polynomials, while being very simple to express, do not readily lend them- selves to direct counting arguments for vertex coloring. As such, it is typical to count the number of colorings of a graph using monomial symmetric polynomials, and then apply a change of basis to express them using elementary symmetric polynomials.

With the core concepts of graphs and graph colorings covered, we still need to understand what incomparability graphs of (3 + 1)-free posets are to understand Stanley’s claim. We first define what a poset is.

Definition 3. A partial ordered set (commonly referred to as a poset) is a binary relation, p, over a set P . This relation must satisfy ≤

Reflexivity: x p x x P • ≤ ∀ ∈

Antisymmetry: if x p y and y p x, then x = y • ≤ ≤

Transitivity: if x p y and y p z, then x p z. • ≤ ≤ ≤

We may isolate a portion of a poset by finding an induced subposet.

ˆ Definition 4. An induced subposet P of a poset (P, p) is a poset whose set is given ≤ by applying an injective function, f, to P . Two elements f(a) and f(b) are comparable in the induced subposet if and only if a and b are comparable in (P, p). ≤

A simple way to consider an induced subposet of a poset (P, p) is to induce the ≤ same equivalence relation on Pˆ P . ⊆ 8 v5 v6 • • v4 vˆ • 4 • v3 vˆ • 3 • v1 v2 vˆ vˆ • • 1 • • 2 P Pˆ

Figure 2.6

Referring to Fig 2.6, we have that P is a poset. We apply the function f : P Pˆ → ˆ where f(vi) =v ˆi for i = 1, 2, 3, 4. For the equivalence relationship of P , we require thatv ˆi is comparable tov ˆj if and only if vi is comparable to vj in P . By definition, Pˆ in Fig 2.6 is an induced subposet of P . The structure of Pˆ is referred to as a (3+1) poset. Any poset which does not contain a (3+1) induced subposet is called (3+1)-free.

For each poset, we can create a graph that describes which nodes of the poset are incomparable. This graph is referred to as the incomparability graph of the poset.

Definition 5. The incomparability graph of a poset P , denoted inc(P ), is the graph whose vertices are nodes in P in which vertices of inc(P ) are adjacent if and only if their corresponding nodes in P are incomparable.

The incomparability graph of Fig 2.6 is the graph shown in Fig 2.7.

Recall Stanley’s e-positivity conjecture states that all (3 + 1)-free posets’ incom- parability graphs are e-positive. That is to say, given some (3 + 1)-free poset, P ,

9 v3 v4 v6 • • •

• • • v1 v2 v5

Figure 2.7 inc(P )’s chromatic polynomial can be expressed as a linear combination of elemen- tary symmetric polynomials with positive coefficients.

10 Chapter 3: Line Graphs

Motivated by Stanley’s e-positivity conjecture, we wish to analyze collections of graphs of graphs which contain all incomparability graphs. The natural choice of graphs would be the collection of claw-free graphs; graphs which do not contain K3,1 (shown in Fig 3.1) as an induced subgraph. Since all (3+1)-free posets’ incomparabil- ity graphs are claw-free [6], we can consider claw-free graphs as a minimal requirement for collections of graphs to study. Ideally, we would find a proper subset of claw-free graphs that contain incomparability graphs as a proper subset. We will attempt to a find a collection of graphs that has enough restrictions to focus our study without being so strict as to simply be the collection of incomparability graphs of (3+1)-free posets. One potential collection of graphs for consideration is the collection of line graphs.

•

•

• • Figure 3.1: K3,1 also known as the claw graph

Line graphs are essentially the answer to the question “How can I turn an edge coloring problem into a vertex coloring problem?” An edge coloring is defined analo- gously to proper vertex coloring. We create a function that assigns a color (or value) to each edge in our graph with the caveat that two edges are not allowed to be colored the same if are incident to each other. We say that two edges are incident if they share a vertex. For example, in Fig 3.2, we see that G1 is not an edge coloring as

11 there are two blue edges incident with each other. G2, on-the-other-hand, is an edge coloring.

• • • •

• • • •

G1 G2

Figure 3.2

Depending on context, it is valid to define edge coloring analogously to vertex coloring in order to impose a notion of proper edge coloring. For the purpose of this study, we do not need this distinction.

As we defined the chromatic polynomial to encode the ways of coloring vertices, we can create an analogous definition of edge chromatic polynomial to encode the possible ways to color the edges.

Definition 6. Let G be a graph and K be the set of all edge coloring functions

κ : E(G) C, The edge chromatic polynomial of G, denoted χˆG is given by: → X Y χˆ(G) = xκ(e). κ∈K e∈E(G)

As stated before, we will only allow a coloring if no two incident edges share a

2 2 color. Hence, the graph G2, from Fig 3.2, will contribute xblackxbluexredxgreen to the edge chromatic polynomial.

We wish to transform any given edge coloring problem into a vertex coloring

12 problem. In order to make this transformation, we will find the line graph of our original graph.

Definition 7. Given a graph, G = (V (G),E(G)), we create the line graph of G, denoted L(G), by letting e V (L(G)) if and only if e E(G), and for every distinct ∈ ∈ pair ei, ej V (L(G)), eiej E(L(G)) if and only if ei and ej are incident in G. ∈ ∈

It is important to note that in this definition we will map multi-graphs to multi-graphs. Hence if two edges are incident twice in G, they must have two edges between them in L(G). Fig 3.3 illustrates this.

e1 e4 • • • e4 e2 • e1 e3 • • e•2 e•3 G L(G)

Figure 3.3

Theorem 3.0.1. χG(x) =χ ˆL(G)

The statement and proof of the previous theorem may be shown in the literature, but we will verify the claim independently.

Proof. By the definition of a line graph, every vertex in L(G) is an edge in G. Thus, coloring vertices in L(G) is equivalent to coloring their corresponding edges in G. Furthermore, since an edge coloring is only valid when incident edges are assigned different colors, our vertex coloring will be valid for L(G) if and only if the edge

13 coloring is valid for G. Thus, the chromatic polynomial of L(G) is equal to the edge chromatic polynomial of G.

We can change an edge coloring problem into a vertex coloring problem via line graphs. Furthermore, we can see from this definition that every graph has a line graph. This mapping, however, fails to be injective. This can be observed by taking any graph G and adding an isolated vertex, v, to the graph. We have that G and (G v ) are not isomorphic, but their line graphs are precisely the same. The added ∪{ } vertex does not contribute any new vertices or edges.

We know that line graphs are a group of simple graphs which allow us to take edge coloring problems to vertex coloring problems, but why should we consider these when attempting to understand e-positivity? First and foremost, the claw graph is not a line graph.

Theorem 3.0.2. The claw graph is not the line graph of any graph.

Proof. To observe this, let us apply a labeling to the claw graph. Without loss of generality, let the claw graph be labeled as shown in Fig 3.4. We may assume for the sake of contradiction that the claw graph is the line graph for some graph G. This implies that e1, e2, e3, and e4 must be the edges of G. Since e1 is adjacent to e2 and e3, but e2 and e3 are not adjacent, we have that e1, e2, e3 must form a path graph with e1 being the middle edge.

e2 e1 e3 • • • •

Attempting to place e4, we will be forced to have e4 incident with either e2 or e3. Thus, the claw graph is not the line graph of any graph.

14 e2 •

e•1 e3 e4 • • Figure 3.4

Since line graphs are claw-free, they satisfy the minimum requirement for any potential collection of graphs to study in order to gain insight on the e-positivity con- jecture. Line graphs are also a well-studied collection of objects. An important result discovered by Whitney is that two connected graphs are isomorphic if and only if their line graphs are isomorphic with one exception; K3 and K3,1 are non-isomorphic graphs whose line graphs are isomorphic [7]. Furthermore through the work of Harary [3], we know which graphs cannot be line graphs. To describe the graphs which can- not be line graphs, we give a list of graphs which cannot be an induced subgraph of any line graph. This list of graphs is referred to as the forbidden graphs for line graphs. The definition of forbidden graphs can change depending on the context, but for our purpose, we will always refer to a graph being forbidden if its existence as an induced subgraph causes a graph to fail be in a desired collection of graphs.

Since the collection of forbidden graphs of line graphs is complete through the work of Harary [3], we can test these forbidden graphs in order to see if they incom- parability graphs of (3+1)-free posets. The collection of forbidden graphs are shown in Fig 3.5.

For each graph in this list, we attempt to construct a poset whose incomparability graph is isomorphic to the given line graph. In Fig 3.6, we show that Gi is an

15 incomparability graphs of the poset Pi for i 1, 2, 3, 4, 5, 6, 7 , whereas, G8 and G9 ∈ { } are are not.

G : G : G : 1 • 2 • 3 • • • • • • • • • • • •

G : G : G : 4 • 5 • 6 • • • • • • • • • • • • • • • •

G : G : G : 7 • • • 8 • • 9 • • • • • • • • • • • • •

Figure 3.5: The 9 Forbidden Line Graphs

P : P : P : 1 • 2 3 • • • • • • • • • • • • •

P : P : P : 4 • 5 6 • • • • • • • • • • • • • • • • •

P : 7 • • • • • •

Figure 3.6: For each i, Pi is the poset whose incomparability graph is given by Gi

For the sake of completion, we show that G8 and G9 are not incomparability graphs of posets.

Theorem 3.0.3. G8 is not the incomparability graph of a poset

16 Proof. Seeking a contradiction, assume that G8 is the incomparability graph of some poset P . Without loss of generality, we may assign a labeling of the vertices of G8 as follows: e c • • a b • •

•f d•

Attempting to construct P , we can place a and b anywhere as they are not com- parable in P . Next, we see that c and d are both comparable to a and b but are not comparable with each other. If c and d are both between a and b, we have that a is comparable to b. If c and d are on opposites sides of either a or b, we have that c is comparable to d. Thus, either both c and d are above or both c and d are below a and b. Without loss of generality, let c and d both be above a and b.

c d • •

a b • •

Now attempt to place e, observe that of the placed elements e is only comparable with d. This implies that e must be below d, else it will be comparable with a and b.

Observe that when placing f, if f cannot be above c without being comparable to a. Furthermore, f cannot be between c and e, otherwise c and e will be comparable. This implies that f must be below both c and e. If f is below e, then we must have f is comparable to d. Thus, we cannot place f.

17 c d • •

a b e • • •

Theorem 3.0.4. G9 is not the incomparability graph of a poset

Proof. Seeking a contradiction, assume that G9 is the incomparability graph for some poset, P . Without loss of generality, let G9 be labeled as shown below.

a • e b • • • f d c • •

Since f is incomparable to all other elements, f must be an isolated node in P . We can arbitrarily place a. Since a is comparable with c and d in P , but c and d are incomparable, we have that both c and d must be both above or both below a. Without loss of generality, let c and d be above a.

c d • •

•a •f

Since b is comparable with d in P , but it is not comparable with a, we have b cannot be placed above d.

18 c d • •

•a b• •f

We observe e is comparable with c but not a. This implies that e is strictly below c. Similarly, e is strictly above b as it is comparable with b but not d. These requirements imply that b must be below c. We then observe that e cannot be in between b and c else they will be comparable. Thus, we cannot place e. G9 is not an incomparability graph.

Thus, we have shown that line graphs are claw free graphs, but they also avoid incomparability graphs of (3+1)-free posets. In particular, we have shown that using line graphs as potential incomparability graphs will avoid the posets that contain P2 through P7. As shown in Chapter 5, this will imply that they avoid all posets that contain any of P2 through P7 as induced subposets.

Since line graphs fail to contain all incomparability graphs of (3+1)-free posets, they are not an ideal choice of study when attempting to gain insight towards the e-positivity conjecture. To use line graphs, we would either need to find further re- strictions on line graphs which would force it then to be a subset of incomparability graphs, or we could focus on line graphs which fail to be e-positive. If there was a clear avenue to the first approach, it would seem more direct to apply those methods towards incomparability graphs in the first place. As for the latter idea, there is no clear indication for what causes a graph to be e-positive or not. Therefore, testing large volumes of line graphs is strictly worse than testing a large volume of incompa- rability graphs. Instead, we propose a modification to the definition of line graphs in

19 an attempt to recover the missing incomparability graphs.

20 Chapter 4: Modified Line Graphs

Recall in the definition of line graph, we have that for every distinct pair ei, ej ∈ V (L(G)), eiej E(L(G)) if and only if ei and ej are incident in G. This definition ∈ implies that if an edge is incident twice in G , then it will induce a multi-edge in L(G) as illustrated in Fig 3.3. A natural modification to this definition would be to to reduce these multi-edge pairs into a single edge between vertices in L(G).

Definition 8. Given a graph G, we construct the modified line graph of G, denoted L˜(G), by first constructing L(G). If L(G) is simple, L˜(G) = L(G). If L(G) is not simple, reduce all multi-edge pairs to a single edge between vertices. This reduction will be L˜(G).

With this change of definition, we need to verify that we still maintain an equiva- lence in the edge chromatic polynomial of G and the chromatic polynomial of L˜(G).

Theorem 4.0.1. χG(x) =χ ˆL(G)(x) =χ ˆL˜(G)(x)

Proof. Since χG(x) =χ ˆL(G)(x), we only need to showχ ˆL(G)(x) =χ ˆL˜(G)(x). Observe that a proper vertex coloring is valid if and only if each pair of adjacent vertices have different colorings. Thus, we may remove or add edges between vertices provided that we do not change the adjacency of vertices. This implies every coloring that is valid in L(G) is valid in L˜(G) and vice versa.

Since the chromatic polynomial is preserved in our modified line graphs, we need to see if we recovered the lost incomparability graphs. To do this, we will refer to list of forbidden graphs of line graphs to see if there exists a graph whose modified line graph is one of the forbidden graphs listed in Fig 3.5. To start, we observe that a vertex with any number of multiple loops in G will map to a single loop in L˜(G). As

21 such, we do not need to consider any loop placements while attempting to find the graphs that map to the forbidden graphs.

Theorem 4.0.2. G1 is not a modified line graph.

v3 •

v1 •

v2 v4 • •

Proof. Seeking a contradiction, assume that G1 is the modified line graph of some graph. Let the largest degree vertex be v1 and let the remaining three vertices be v2, v3, and v4. Observe, v1 is incident to v2 and v3, but v2 and v3 are not incident to each other. This implies that v1, v2, and v3 form the following graph.

v2 v1 v3 • • • •

We observe when attempting to place v4, it must be incident to v1, but it cannot be incident to v2 or v3. Thus, there is no place for v4. G1 is not a modified line graph.

Theorem 4.0.3. G7 is not a modified line graph.

v1 v2 v3 • • •

v•6 v•5 v•4

Proof. Assume that G7 is the modified line graph of some graph. Label the vertices of G7 in clockwise order starting in the upper-left-hand corner v1 through v6.

22 Observing that v1, v2, and v6 are all incident with each other, and v3 is only incident with v2, we have two possible cases: either v1 and v6 have two shared vertices, or they have one shared vertex. Case 1: Two Shared Vertices

v1 v2 v3 • • • • v6

In case 1, we see that v5 must be incident with v6, but cannot be incident to v1. Thus, there is no valid placement.

Case 2: One Shared Vertex

•v6 v1 v2 v3 • • • •

Since v4 is incident to v3 but not to any of the edges already placed, we are forced to place v4 such that it is connected to the vertex of v3 which is not shared by v2.

•v6 v1 v2 v3 v4 • • • • •

Now observe, v5 must be adjacent to v2, v3, v4, and v6. We are forced to place v5 as the edge between v3 and v4’s common vertex and v2 and v6’s common vertex. This now makes v5 incident with v1 which contradicts our original graph. Thus, G7 is not a modified line graph.

Theorem 4.0.4. G9 is not a modified line graph.

23 v6 • v5 v2 • • v•1 v4 v3 • •

Proof. Label the vertex with highest degree v1 and the remaining vertices v2 through v6 clockwise.

We observe when placing v1, v2, and v4 that v1 is incident to v2 and v4, but v2 and v4 are not incident to each other. This implies they create a path graph with 3 edges. When placing v3, there are three possibilities. In each case, we will have the path graph, P3, with edges labeled v1, v2, v4 where v1 is the middle edge. For the first two placements (illustrated below by the red dashed line), we will have that v3 has the vertex shared by v1 and v4 as one of its endpoints. In this case, we will be forced to make v5 incident with v2 or v3.

• v2 • v1 • v4 •

Hence, the only valid placement for v3 is where it does not share a vertex with the common vertex of v1 and v4. Attempting to place v5, we see that it is incident to v1 and v4, but is not incident to v2 or v3. Therefore, v5 can only be placed on the shared vertex between v1 and v4.

v3

v2 v1 v4 • • • • v5 •

We cannot have v6 share v1 and v5’s common vertex, otherwise, v6 will be incident

24 with v4. We observe that v6 may not share v1 and v2’s common vertex otherwise v6 is incident to v3. Thus, there is no way to make v6 incident with v1 that does not make it incident with v3 or v4. G9 is not a modified line graph.

Theorem 4.0.5. Graphs G2 through G6 and G8 in Fig. 3.5 are all modified line graphs.

• • •

•

• • • • • • • • • 0 0 0 G2 G3 G4

• • • •

• • • • • • • •

• • • • • 0 0 0 G5 G6 G8

ˆ 0 Figure 4.1: For i = 1, 2, 3, 4, 5, 6, 8, L(Gi) = Gi.

In conclusion, modifying the definition of line graph recovers all of the forbidden graphs of line graphs with two exceptions. Modified line graphs are claw free; a de- sired result, however, modified line graphs do not contain G7 from Fig 3.5. Since G7 has been shown to be an incomparability graph of a poset, studying modified line graphs will not help gain insight towards the e-positivity conjecture.

As previously discussed line graphs do not seem to be an ideal approach when attempting to gain insight on the e-positivity conjecture. For the same reason, modi-

25 fied line graphs are also not ideal. The issues of studying line graphs are worsened as the change in definition also comes at the cost of Whitney’s Isomorphism Theorem. What line graphs and modified line graphs give us, is a different approach towards. Namely, we can attempt to find the forbidden graphs of incomparability graphs.

26 Chapter 5: Incomparability Graphs

We may attempt to create a list of forbidden graphs for the incomparability graph of posets. In order to do so, we must first prove that the induced subgraph of an incomparability graph is the induced subposet of the corresponding poset. This result is not original, and the proof is included only for the sake of completion.

Theorem 5.0.1. Let P be a poset with incomparability graph inc(P ). A poset Q is an induced subposet of P if and only if inc(Q) is an induced subgraph of inc(P ).

ˆ Proof. Let P = (X, p) be a poset and Q = (X, p) be an induced subposet of P . ≤ ≤ Let inc(Q) be the incomparability graph of Q. Observe,v ˆ V (inc(Q)), if and only ∈ if there exists a corresponding vˆ∗ Xˆ by definition of incomparability graph. By ∈ definition of induced subposet, there exists a corresponding v∗ X. As v∗ is an ∈ element of P , there exists a v V (inc(P )). ∈

∗ ∗ Furthermore for eachv ˆivˆj E(inc(Q)), the correspondingv ˆ andv ˆ are not com- ∈ i j parable in Q by definition. Since Q inherits its partial ordering from P , we have

∗ ∗ the corresponding vi and vj are not comparable in P . This implies that there is the corresponding vi, vj V (inc(P )) are adjacent. Hence, vivj E(inc(P )). This implies ∈ ∈ that inc(Q) is an induced subgraph of inc(P )

Assume P = (X, p) is a poset with a given incomparability graph, inc(P ). Let ≤ G be an induced subgraph of inc(P ). By definition, V (G) V (inc(P )) which in ⊆ turns implies that an element of V (G) has a corresponding node in P . Let Xˆ be the subcollection of nodes of P that have corresponding vertices in V (G). Observe, vivj E(G) implies that vivj E(inc(P )). By definition of incomparability graph, ∈ ∈ 27 ˆ vi is not comparable to vj in P . Therefore, the poset created by (X, p) is an induced ≤ subposet of P .

Corollary 5.0.1.1. Let G be a graph and H be an induced subgraph of G. If H is not an incomparability graph, then G is not an incomparability graph.

Therefore, inducing a subgraph of the incomparability graph is equivalent to in- ducing a subposet and vice versa. This result and its following corollary allow us to start building a list of forbidden graphs.

Before we begin analyzing potential forbidden graphs, we will first observe two commonly used techniques for developing a poset from the forbidden graph. We may observe from the definition of incomparability graph, that an edge will exist between two vertices if and only if they are incomparable in the poset. This implies that if there exists a vertex that is adjacent to all other vertices, then the vertex must exist as an isolated node in the poset. As an immediate application, we can observe that any complete graph, Kn, must be the incomparability graph of a poset with n nodes where no elements are comparable. This method also implies that when observing a potential incomparability graph, we should start by removing all of the vertices which are adjacent to all other vertices.

Another commonly used technique to developing the poset from the incomparabil- ity graph is a “stacking” method. We may observe that if an incomparability graph has more than one connected component, then we may consider each component in- dividually to create the induced subposet for that component. Since every element in one component is comparable to every element in the other components, we may take the induced subposets created by these components, stack them on top of each other, and connect the vertices accordingly. An example of this method is given in

28 Fig 5.1. This method implies that when attempting to find the forbidden graphs, we can consider only simple, connected graphs as any graph with two or more compo- nents will only fail to be an incomparability graph if one of the components fail to be an incomparability graph.

•

• •

• • • • • • • •

• • •

• • −1 P4 K1,3 I (P4 K1,3) ∪ ∪ Figure 5.1

A common collection of graphs is the set of all path graphs. A path graph is essen- tially a graph that forms a straight line. We denote the path graph on n vertices by Pn.

Theorem 5.0.2. Each path graph is the incomparability graph of some poset.

Proof. Let Pn be a path graph with n vertices. Let v1 be a leaf of Pn, v2 the vertex adjacent to v1, v3 the other vertex adjacent to v2, and so on. We claim for n 5, Pn ≥ is the incomparability graph of a poset of form shown in Fig 5.2.

We first consider P5 as a potential incomparability graph. We observe that v2 is not comparable to v1 or v3, but v1 and v3 are comparable. Without loss of generality let v3 be above v1. Since v4 is comparable to v2 and is not comparable to v3, we have that v4 must be placed above v2.

29 vk−1 vk • •

. .

v7 v8 • •

v5 v6 • •

v3 v4 • •

v1 v2 • • Figure 5.2

Finally, v5 is not comparable to v4. This implies that v5 cannot be placed below v1 or v2. Furthermore since v2 and v3 are not comparable, v5 cannot be placed in- between v2 and v3. Therefore, v5 must be placed above v3.

v5 •

v3 v4 • •

v1 v2 • •

Assume that the structure shown in Fig 5.2 is the poset for an incomparability

30 graph of the form of a path graph for some k 1 number of vertices. Observe that − for Pk, we know the structure of of the poset for its first k 1 elements. Focusing on − k+2 vk, we observe that if vk is placed between vk−2i and vk−2i−2 for 1 i , we will ≤ ≤ 2 have that vk−2i is comparable to vk−2i−1 if k is odd or vk−2i is comparable to vk−2i+1 if k is even. By our labeling, we have that consecutive vertices are not comparable.

Thus, vk cannot be placed between any of these vertices. Furthermore if vk is placed below v1 and v2, we will have that vk is comparable to vk−1. Thus, the only valid placement for vk is above vk−2.

Since we know the form of the poset of all path graphs, we can now look at all cycle graphs.

Theorem 5.0.3. Cn is not an incomparability graph of a poset for n 5. ≥

Proof. Let Cn be a cycle graph whose vertices are labeled v1, v2, . . . , vn counterclock- wise from any given starting vertex with n 5. We have Cn vn = Pn−1. By the ≥ \ previous argument, we can construct the poset whose incomparability graph is Pn−1.

Attempting to bring back vn, we have from the previous argument that vn must be placed above vn−2. Observe, placing vn above vn−2 implies that vn is comparable to v1. Thus, there are no valid placements for vn. Cn is not the incomparability graph of any poset.

Corollary 5.0.3.1. Wn is not an incomparability graph of a poset for n 6. ≥

Proof. We simply observe the central vertex is incomparable to all other vertices.

Thus, Wn is an incomparability graph if and only if Cn is an incomparability graph.

31 We may also observe that complete graphs are the incomparability graph of posets in which no element is comparable, graphs with no edges are incomparability graphs of totally ordered graphs; graphs in which any two given elements must be comparable, and bipartite graphs of the form Km,n are the incomparability graphs of (m + n)- posets.

m dots n dots m dots n dots

• • • •

• • • • ......

• • • •

• • • •

• • • • P inc(P )

Figure 5.3: An (m + n)-poset, P , and its incomparability graph, inc(P ).

Finding common families of graphs that are not incomparability graphs is harder. In order to find graphs which fail to be incomparability graphs, we will start by ob- serving all simple, connected graphs. Richard Stanley checked all possible graphs with 7 or less vertices through the use of technology [6], but as described above, that is unnecessary. Restricting simple, connected graphs to 5 or fewer vertices [8], we find that only C5 fails to be an incomparability graph. Allowing for 6 vertices [1], we find that there are three additional graphs which fail to be incomparability graphs. Of these graphs, only one has been named: the 3-Sun graph. One may recall from the introduction that this is an example of a graph which is claw free, but is not e-positive.

32 Theorem 5.0.4. The 3-sun graph (Fig 1.1) is not an incomparability graph of a poset.

Proof. Assume that the 3-Sun graph is the incomparability graph of some poset P . Let the 3-Sun graph be labeled as follows.

v5 •

v2 •

v•4 v•1 v•3 v•6

Vertices v1, v2, and v3 are not comparable in P . Observe, v4 is comparable to v2 and v3. This implies that v2 and v3 are both above or both below v4. Without loss of generality, let v2 and v3 both be below v4.

v4 •

v•1 v•2 v•3

Similarly, observe that v6 is comparable with v1, v2, and v4, but not v3. This forces v6 to be below both v1 and v2.

v4 •

v•1 v•2 v•3

v•6

33 Note: The heights shown in the above diagram are not fixed. We only have v6 is below v1 and v2 and v4 is above v2 and v3.

Attempting to place v5, we first observe that v5 cannot be above v4 or below v6 else v5 is comparable to v2. Since v1 and v3 are not comparable to each other, we have that v5 cannot be in-between them. We can observe that if v1 is higher than v4, we cannot place v5.

v•1 v4 •

v•2 v•3

v•6

In a similar manner, v3 cannot be placed below v6 else we can place v5. This implies that v1 and v3 are between v4 and v6. This placement also fails to produce a valid location for v5. Thus, the 3-Sun graph is not an incomparability graph.

Theorem 5.0.5. Neither of the graphs in Fig 5.4 are incomparability graphs of a poset.

• • •

• • • •

• • • • • A B

Figure 5.4

Proof. Starting with graph A in Fig 5.4, label the vertices as shown in Fig 5.5.

34 v6 v1 • •

v2 v5 • •

v•4 v•3

Figure 5.5

We can observe that v1, v2, v3, v4 create a path. Without loss of generality, we will let v1 be below v3 and v4. Noting that v5 is only comparable to v2, we have the following:

v3 v4 v5 • • •

v•1 v•2

We observe that if v6 is above v4, then v6 will be comparable to v1. Similarly if v6 is below v2, then v6 will be comparable to v5. This implies that v6 must be in-between v2 and v4. We have two cases to consider. If v3 is below v6, we will have that v3 is comparable to v4. If v3 is above v6, then v2 will be comparable to v3. In either case, we there is a contradiction. Thus, we have that graph A in Fig 5.4 is not an incomparability graph.

Considering graph B in Fig 5.4, we will label the vertices as shown in Fig 5.6.

35 v6 •

v4 v5 • •

v•1 v•2 v•3

Figure 5.6

Focusing on v1 through v4, observe that since v3 is comparable to v1 and v4 it must be above both or below both. Without loss of generality, we will assume that v3 is above v1 and v4. Since v5 is only comparable to v1, it is forced to be above v1.

v3 v5 • •

v•4 v•1 v•2

Note that v6 cannot be above v3, and it cannot be below v1 otherwise v6 will be comparable to v4 or v5 respectively. Thus, v6 must be in-between v1 and v3. If v2 is below v6, then v2 will be comparable to v3. If v2 is above v6. then v2 will be comparable to v1. Thus, v6 cannot be in-between v1 and v3. This implies that there are no valid placements for v6. Hence, graph B is not an incomparability graph.

By exhausting all 143 simple, connected graphs with six or less vertices, we find that there are only twelve graphs which fail to be the incomparability graph of some poset. Eleven of these graphs are highlighted in Appendix A; the other is C5. Fur- thermore, we observe that seven of the graphs with six vertices contain C5 as an induced subgraph. Thus, there are only five forbidden graphs with six or less vertices

36 for incomparability graph. These graphs are C5, C6, the 3-Sun, and the two graphs from 5.4.

37 Chapter 6: Future Work

We have started developing a list of forbidden graphs for incomparability graphs by finding the five forbidden graphs with six or fewer vertices. There are two options for continuing at this point. We can either continue searching for the list of forbidden graphs of incomparability graphs, or we can look only for the forbidden graphs of the incomparability graphs of (3+1)-free posets. The former would be the ideal approach. If we were able to establish the list forbidden graphs of incomparability graphs then as a corollary, we would have the forbidden graphs of incomparability graphs of (3+1)- free posets. This corollary follows as (3+1)-free posets’ incomparability graphs are claw-free. That is to say, the claw is a forbidden graph of this family of graphs.

On-the-other-hand, we may choose instead to start with the claw graph being a forbidden graph. This approach would be simpler for finding forbidden graphs with the use of technology, as it would allow for the study of smaller number of graphs. As an example, there are only 50 claw-free, simple, connected graphs on 6 vertices (A022562 OEIS). In contrast when allowing the claw graph to exist as an induced subgraph, there were 112 graphs to study.

38 Bibliography

[1] DragoˇsCvetkovi´cand Milenko Petri´c.A Table of Connected Graphs on Six Vertices. Discrete Mathematics. 50 (1984), 37-49.

[2] Vesselin Gasharov, Incomparability Graphs of (3+1)-Free Posets are s-Positive. Discrete Mathematics. 157 (1996), 193-197.

[3] Frank Harary, Graph Theory. Addison-Wesley Publishing Company (1969).

[4] Jeffrey C. Lagarias. The 3x + 1 Problem and its Generalizations. The American Mathematical Monthly 92 (1985), 3-23

[5] Rosa Orellana and Geoffrey Scott, Graphs with Equal Chromatic Symmetric Functions. Discrete Mathematics. 320 (2014), 1-14

[6] R.P. Stanley, A Symmetric Function Generalization of the Chromatic Polynomial of a Graph. Adv. Math. 111 (1995), 166-194.

[7] Hassler Whitney. Congruent Graphs and the Connectivity of Graphs. American Journal of Mathematics. 54 (1932), 150–168.

[8] http : //www.graphclasses.org/smallgraphs

39 Appendix A: Connected Simple Graphs with 6 Vertices

In the following table [1], we show all of the connected, simple graphs with 6 ver- tices. Since C5 is the only simple connected graph with 5 or less vertices that fails to be an incomparability graph, we will highlight vertices of degree 5 with a yellow dots, and will use purple lines to highlight if C5 is an induced subgraph. We can note that any graph which contains C5 as an induced subgraph will fail to be an incomparabil- ity graph, and any simple graph with 6 vertices that contains at least one vertex of degree 5 must be an incomparbility graph of some poset unless it is W6. The wheel graph fails as it also contains C5 as an induced subgraph.

Furthermore, green check marks are given to graphs which are incomparability graphs of some poset (the poset is drawn near the graph in most cases), red x’s are given to graphs which are not incomparability graphs because they contain C5 as an induced subgraph, and red boxes around graphs are drawn to highlight forbidden graphs of incomparability graphs.

40 41 42 43 Appendix B: Edge Chromatic Polynomial Calculator

44 45 46 Curriculum Vitae :

DAVID NICHOLS

1712 Brookwood Dr., Winston-Salem, NC 27106 (509)429-9503 [email protected]  OBJECTIVE Seeking a PhD in Mathematics with a focus in combinatorics.

EDUCATION Wake Forest University, Winston-Salem NC August 2018 - Present M.S. Mathematics Eastern Washington University, Cheney WA April 2012 - June 2017 B.S. Mathematics

RESEARCH Wake Forest University (Summer 2019 - Spring 2020) Master’s thesis studying Richard Stanley’s e-positivity conjecture. Advised by Dr. Sarah Mason. Combinatorial Game Theory (Spring 2017) Undergraduate thesis studying combinatorial game theory. Advised by Dr. Dale Garraway.

WORK EXPERIENCE Wake Forest University August 2018 - Present Teaching Assistant Full Teaching Assistant-ship · Teaching Assistant for Calc I (Fall 2019), Calc II (Fall 2018), and Calc III (Spring 2019). · Responsibilities include one-on-one tutoring, leading group study sessions, proctoring exams, and grad- · ing assignments. Wilson Creek High School August 2017 - August 2018 Middle School and High School Math Teacher Middle school teacher and high school teacher covering 6th through 11th grade mathematics. · Developed strong classroom preparation and management skills. · Eastern Washington University August 2015 - June 2016 Tutor Tutored undergraduate mathematics courses at Eastern Washington University. · Courses included pre-university mathematics, calculus series, differential equations, linear algebra, etc. ·

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