CHROMATIC FOR GRAPHS WITH VERTICES

Sarah E Adams

A Thesis

Submitted to the Graduate College of Bowling Green State University in partial fulfillment of the requirements for the 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 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 . 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 ...... 7 2.1 Definitions and Examples ...... 8 2.2 Coloring Algorithms ...... 11 2.3 Greedy Algorithm Examples ...... 12 2.4 The ...... 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 ...... 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 = {1, 2, 3, 4, 5, 6} and edges E = {a = (1, 2), b = (1, 5), c = (2, 3), d = (2, 5), e = (3, 4), f = (4, 5), g = (4, 6)}. 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. Also, we do not allow loops, as stated in Definition 1.1.4. Hence, we 4 will only be working with simple graphs.

1.2 Common Families of Graphs

We know from Gross et al. (2014), there are many different types of simple graphs. In this section, we will present a few common examples of simple graphs.

Definition 1.2.1. The Empty graph En with n vertices have n isolated vertices with no edges. The empty graph with 1 node is called a singleton graph. See Figure 1.5 for the empty graph with three vertices.

1

2 3

Figure 1.5 An example of an empty graph with 3 nodes

Definition 1.2.2. The Kn with n vertices is the graph where each vertex is con- nected to all the other vertices of the graph. See Figure 1.6 for K3 and K4

1

2 3

1 2

3 4

Figure 1.6 The graphs K3 and K4 5 Definition 1.2.3. A bipartite graph G is a graph whose vertex set V can be partitioned into two subsets U and W, such that each edge of G has one endpoint in U and on endpoint in W. The pair

U, W is called a bipartition of G, and U and W are called the bipartition subsets

Example 1.2.4. The bipartite graph shown in Figure 1.7 shows the bipartition subsets indicated by the labels A and B.

A1 B1

A2 B2

A3

Figure 1.7 A bipartite graph

Definition 1.2.5. The Pn is a graph with n vertices that has two vertices of degree 1, and the other n − 2 vertices of degree 2. A path graph is therefore a graph that can be drawn so that all of its vertices and edges lie on a single straight line Gross et al. (2014). See Figure 3.2 for the P4 graph.

1 2 3 4

Figure 1.8 The P4 graph

Definition 1.2.6. The graph Sn of order n, (sometimes known as an ”n-star” from Sweet, Li, Cheng, Liptk, and Steffy (2018)), is a graph with n vertices where one vertex has degree n − 1 and

the other n − 1 vertices have degree 1. See Figure 1.9 for the S6 graph. 6 5

4 6 1

2 3

Figure 1.9 The S6 graph

Definition 1.2.7. Let n ≥ 3, a , Cn is a simple graph with n vertices and n edges that can be drawn so that all of its vertices and edges lie on a single circle. Gross et al. (2014). See

Figure 1.10 for the graph C4

2

1 3

4

Figure 1.10 The C4 graph 7

CHAPTER 2 GRAPH COLORING

Graph Coloring began as people were interested in coloring maps. People were interested in this idea because they wanted to find the minimum number of colors necessary for coloring various configurations such as maps. Maps can be represented as graphs by labeling each of the countries with a vertex, and then connecting each of the neighboring countries with an edge. For example, suppose you wanted to color the map of South America. We want to color South America in a way so that each of the countries that share a border are colored with different colors. The question arising, what is the minimum number of ways to color this map? To begin to answer this question, we must first translate the map from Figure 2.1 into a graph configuration, as shown in Figure 2.2.

Figure 2.1 South America Mapping 8 V Co G S F

E

B P

Bo

Pa

C A U

Figure 2.2 The graph of South America

The idea of graph coloring is to assign each vertex a color so that no two adjacent vertices have the same color. For maps this translates to coloring all the countries so that no two neighboring countries have the same color.

2.1 Definitions and Examples

We will start by giving a few definitions from Wilson and Beineke (1979) and Lewis (2016).

Definition 2.1.1. (λ -Coloring, λ -Colorable). Let G be a graph and let λ ∈ N. A proper λ- coloring of a simple graph G = (V,E) is an assignment of colors {c1, c2, c3, ..., cλ} to the vertices of G so that no two adjacent vertices have the same color. In other words, a λ-coloring is a function f : V (G) → C = {c1, c2, ..., cλ} such that if e ∈ E(G) and v, w are the vertices of e than f(v) 6= f(w). A graph that has a λ-coloring is said to be λ-colorable.

Example 2.1.2. Let’s take an example. Consider the path graph, P5 as seen in Figure 2.3(a). The goal is to color the graph so that no two adjacent vertices are the same color. We will label the vertex 1 as blue. Continuing this process, since vertex 2 is connected to vertex one, we can not color it blue, so we color it yellow. Continuing to vertex 3, since it is adjacent to the yellow vertex, 9 we can color this vertex anything except for yellow. We can color it blue again. By continuing this process we can label vertex 4 and 5 as yellow and blue, respectively. At the end we get the coloring in Figure 2.3(b).

a) 1 2 3 4 5

b)

Figure 2.3 Proper Coloring of P5 graph

Example 2.1.3. Let K5 be the complete graph with 5 vertices. Since it is a complete graph, every vertex is connected to all the other vertices with an edge. Since each vertex is adjacent with all of the other vertices, each vertex will be forced to be a different color. So we need a minimum of 5 colors to properly color this graph. An example of such coloring is shown is Figure 2.4.

Figure 2.4 Proper Coloring of K5 graph

From our previous example, we can see that there is a minimum of 5 colors needed to properly color the graph K5. This becomes quite obvious as you begin to color. However, coloring graphs can be cumbersome without a strategic method for the coloring process. This leads us to trying to find a method to calculating the minimum numbers of colors needed to color a graph. We define this as the chromatic number. 10

Figure 2.5 Proper Coloring of an empty graph

Definition 2.1.4. We define chromatic number of a graph G as the minimum number of colors needed to color its vertices so that no two adjacent vertices are assigned the same color. This is

often denoted by as χ(G). If χ(G) = k, we say that G is k-chromatic. If χ(G) ≤ k, then G is k-colourable.

Example 2.1.5. As discussed in chapter 1, the empty graph consists of isolated vertices. This means that each of the n vertices can be colored the same color. Since this is the case, the chromatic number of an empty graph is 1. An empty graph with three vertices is shown in Figure 2.5.

Example 2.1.6. In Figure 2.4, we looked at the graph K5, and saw that the graph was 5-chromatic.

Consider the graph Kn, notice that each vertices must be a different color and there are n-vertices

so, we must have that Kn is n-chromatic.

Example 2.1.7. The path graph Pn(n ≥ 2) is 2-chromatic since, you can properly color the graph by alternating the colors. You can see this by generalizing the coloring in Figure 2.3.

Example 2.1.8. The cycle graph Cn is either 2-chromatic or 3-chromatic depending on n, the number of vertices. The simplest case being n = 3, then one can see that we need three colors to properly color the graph. Continuing to n = 4, one can see that the chromatic number is two. This is the same for all even n. Similarly for each odd n, one can see that we need a minimum of 3 colors to color the graph. An example of the proper coloring of a C5 graph is shown in Figure 2.6.

Theorem 2.1.9. If H is a subgraph of G, then χ(H) ≤ χ(G). 11

Figure 2.6 Proper Coloring of an C5 graph

Proof. Since H is a subgraph of G, the vertices of H are a subset of the vertices in G, and the edges of H are a subgraph of the edges of G. Any coloring of G provides a proper coloring of H, simply by assigning the same colors to vertices of H that they have in G. This means that H can be colored with χ(G) colors, perhaps even fewer, which is exactly what we want to prove. See Guichard (2020).

2.2 Coloring Algorithms

Coloring problems have been studied for over a century. It may be quite challenging to deter- mine the minimum number of colors needed for coloring a graph. Thus, it is helpful to develop methods to color graphs. One method to coloring graphs is the greedy coloring algorithm as stated in Janczewski (2009). Although not always successful in finding χ(G), the greedy coloring algorithm, is a notable method for determining proper coloring of a graph with as few colors as possible. Because of the occasional inconsistency of finding the chromatic number, we refer to the algorithm as greedy. Despite the greedy nature, the algorithm provides some sort of method for proper coloring. To

proceed, we first need to number each of the colors consecutively: {c1, c2, c3...}. The procedure is as follows:

1. Give an ordering for the vertices of the graph G say {v1, v2, v3, ..., vn}.

2. Color the first vertex v1 with color c1, set i = 1. 12

3. Take the next vertex vi+1. Color it with the lowest-numbered color that has not been used on

any previously-colored vertex adjacent to vi+1. Change i to i + 1

4. Repeat the previous step until all vertices are colored.

If we have enough colors this algorithm will provide us with a proper coloring for G. Notice that the amount of colors needed depends on the ordering we give for the vertices of G.

2.3 Greedy Algorithm Examples

Example 2.3.1. To apply the greedy coloring algorithm, let’s begin with a K3 graph as shown in

2.7. We will first assign the order of the vertices by labeling the three vertices as {v1, v2, v3}. Say

we have 4 color options, namely, {c1 = red, c2 = blue, c3 = yellow, c4 = green}. We will color

v1 with c1. Next we color v2 with c2 because it is the lowest numbered color that has not been used

on any previously-colored vertices adjacent to v2. Finally, v3 must be c3 because it is the lowest

numbered color that has not been used on any previously-colored vertices adjacent to v3.

2 1

3

Figure 2.7 Greedy Coloring of K3 graph

Example 2.3.2. Say that we want to color the graph G shown in Figure 2.8. If we consider the

order v0, v1, v2, v3, v4 (as shown in Figure 2.8(a))and we have colors c1 = yellow, c2 = blue,

c3 = pink, c4 = green, we can begin to assign colors to each of the vertices. We write a function

C : {v0, v1, v2, v3, v4} → {c1, c2, c3, c4}. We start with C(v0) = c1 = yellow. Next, we move to v1.

Since v1 is adjacent to v0, we move to the next color. Thus, C(v1) = c2 = blue. Continuing to v2, we must have C(v2) = c3 = pink because v2 is adjacent to both v0 and v1. Next, C(v3) = c1 =

yellow because c1 is the lowest-numbered color that has not been used on any previously-colored 13 vertices adjacent to v3. Finally, we have that C(v4) = c2 = blue since v4 is only adjacent to v3. And so we can color the graph G using 3 colors.

Example 2.3.3. As we have stated previously, the number of colors used depends on the order in which vertices are processed. We consider the order v0, v1, v2, v3, v4 as shown in Figure 2.9(a).

Let us have colors c1 = yellow, c2 = blue, c3 = pink, c4 = green, we can begin to assign colors to each of the vertices. We want a function d : {v0, v1, v2, v3, v4} → {c1, c2, c3, c4}. We

start with d(v0) = c1 = yellow. Next, we move to v1. Since v1 is adjacent to v0, we move to

the next color. Thus, d(v1) = c2 = blue. Continuing to v2, we must have d(v2) = c3 = pink

because v2 is adjacent to both v0 and v1. Next, d(v3) = c1 = yellow because c1 is the lowest-

numbered color that has not been used on any previously-colored vertices to v3. Finally, we have

that d(v4) = c4 = green since v4 is adjacent to v1, v2, and v3. So for the ordering in Figure 2.9(b) we need four colors to color the graph G.

1 0

3 4 2 a)

1 0

3 4 2 b) Figure 2.8 Greedy Coloring of Graph G using three colors

To see another example of how the greedy algorithm can sometimes not give us the minimum number of colors, let’s look at the next example shown is Figure 2.10 and Figure 2.11.

Example 2.3.4. Suppose we want to color a graph using the greedy coloring method with ordering 14

1 0

4 3 2 a)

1 0

4 3 2 b) Figure 2.9 Greedy Coloring of Graph G using four colors

shown in Figure 2.10(a). We order the colors as c1 = Blue, c2 = Green, c3 = Orange, c4 = Red. We will color the graph in Figure 2.10(a) using the greedy coloring method so we begin by letting v1 = c1. Following the algorithm, we get the coloring shown in Figure 2.10(b). And so we only need 2 colors.

Example 2.3.5. Suppose we want to do the same for the graph in Figure 2.11(a). Consider the order of the vertices of the graph shown in 2.11(a), and the ordering of colors c1 = Blue, c2 =

Green, c3 = Orange, c4 = Red. We will color the figure using the greedy coloring method. To color the graph in Figure 2.11(a) using the greedy coloring method, we begin by coloring the vertex 1. Similarly, following the algorithm, we get the coloring shown in Figure 2.11(b). This time we needed 4 colors.

Overall, by diligently coloring and avoiding missteps, we can produce a proper coloring. It is hard to determine the amount of colors that will be needed because it depends in what order the vertices are colored. Nevertheless, the greedy coloring method has been proved to be a resourceful algorithm when coloring.

Theorem 2.3.6. (Greedy Coloring Theorem) If d is the largest of the degrees of the vertices in a 15

1 2

3 4

a) 5 6

B G

G B

B G b) Figure 2.10 Another example of greedy coloring

graph G, then G has a proper coloring with d + 1 or fewer colors, i.e., the chromatic number of G is at most d + 1. See Guichard (2020).

The degree of a vertex indicates the number of edges that are incident with the vertex. This gives an upper bound on the chromatic number, but the real chromatic number may be below this upper bound. For example, suppose the highest degree of a vertex is d = 6, so the Greedy Coloring Theorem states that the chromatic number is no more than 7. The following example shows that this is not always a sharp estimate for χ(G).

Example 2.3.7. Suppose we have the S5 graph in Figure 2.12. We know that the largest degree of the vertices is d = 4. In particular, we see vertex 5 in Figure 2.12 has d = 4. So from Theorem

2.3.6, the graph S5 has a proper coloring with 5 or fewer colors. However, notice that S5 can be colored using only 2 colors. Thus, χ(S5) = 2 16

1 3

5 6

a) 4 2

B G

O R

G B b) Figure 2.11 Another example of greedy coloring

4

3 5 1

2

Figure 2.12 The S5 graph

2.4 The Four Color Theorem

So how does this relate to coloring maps, and is it possible to properly color every map with only four colors? According to Alfeld (1996) this question came to fruition when a student of Augustus De Morgan (1833-1886), named Frederick Guthrie (1833-1886) discussed with his pro- fessor a mathematical discovery of his older brother at University College London on 23 October 17 1852. While coloring the counties of a map of England, Francis Guthrie (1831-1899) observed that he could color them using four colors. Eventually this led to the conjecture that the regions of every map can be colored with four or fewer colors in such a way that every two regions sharing a common boundary are colored differently. The first printed reference for the conjecture is due to Cayley in 1878.

Theorem 2.4.1. The Four Color Theorem Any map in a plane can be colored using four-colors in such a way that regions sharing a common boundary (other than a single point) do not have the same color.

Figure 2.13 United States Map using Four Colors

According to Robertson, Sanders, Seymour, and Thomas (1997) and Leward (2014), there were many contributions to the Four Color Problem. Kempe and Tait, both had failed attempts at solving the problem, but formulated theorems that proved to be helpful steps towards the solution. Their contributions include Kempe chains, and 3-edge-coloring. Birkhoff contributed greatly to the four color theorem by considering rings of faces in graphs. Birkhoff’s technique was to remove a ring of faces and every face inside the ring. You start by coloring everything on the outside of the ring, then color the faces inside the ring, and the ring again (after possibly recoloring). Finally, you produce the whole map. Later on in his study, Birkhoff introduced the chromatic polynomial. Birkhoff’s work allowed Franklin to prove the 18 four color conjecture in 1922 for maps with at most 25 regions. With all of these contributions, mathematicians continued to develop progress on the four color problem. In the end, this theorem was proved by Appel and Haken in 1976 Appel and Haken (1977), when they published their proof of the Four Color Theorem assisted by computers as stated in Robertson et al. (1997). 19

CHAPTER 3 CHROMATIC POLYNOMIALS

3.1 History of the Chromatic Polynomial

According to Leward (2014), the chromatic polynomial, was introduced by American math- ematician George David Birkhoff in 1912 after being inspired by the four-color problem. While attempting to solve the four-color problem, Birkhoff introduced the chromatic polynomial. By looking at maps as simple graphs, and defining proper coloring of a graph, Birkhoff hoped to prove that any two dimensional map can be colored with just four colors so that none of the neigh- boring bodies have the same color. Although, he failed at solving the four-color problem, his work influenced many. Birkhoff noticed that the number of ways to color a map using λ colors can be written as a polynomial that depends on λ. In other terms, we can define a polynomial function P (G, λ) that gives us the number of proper colorings of the graph G with λ colors. The chromatic polynomials became an important object in algebraic and continue to peak the interest of many mathematicians today. Although Birkhoff’s definition was limited to planar graphs, the concept of chromatic polynomials continues to grow. Currently, chromatic polynomials encapsulate graphs of a great variety. For this discussion, we will focus on defining chromatic polynomials and applying the definition of chromatic polynomials for simple graphs, (ie. finite graphs without loops or multiple edges.) The results in this chapter are taken from Wilson and Beineke (1979) and Guichard (2020).

3.2 Defining Chromatic Polynomials

Definition 3.2.1. Let G be a graph. The chromatic polynomial P (G, x) ∈ Z[x] of G is determined by P (G, λ) = the number of distinct proper λ-colorings of G for all λ ∈ N. Two λ-colorings c

0 and c’ of G from the same set {c1, c2, c3, ...cλ} of colors are considered different if c(v) 6= c (v) for some vertex v of G.

Recall, we defined proper coloring as an assignment of colors to the vertices of a graph so that no two adjacent vertices have the same color. First, let’s discuss the meaning of the definition 20 in terms of λ coloring. The goal is to be able to write a polynomial function that expresses the number of proper colorings. That is, if we take λ = 2 and P (G, 2) = 8, we are implying that with two colors, the graph can be colored in eight different ways. Additionally, if we say λ = 2 and we discover that P (G, 2) = 0, we are saying that there are zero ways to color the graph with two colors. Now that we understand the meaning of the definition, we will begin to apply what we know to common simple graphs.

3.3 Chromatic Polynomials of Simple Graphs

Example 3.3.1. In Figure 3.1, we have an empty graph, E3 with three vertices. One can show that

3 its chromatic polynomial is P (E3, λ) = λ . Indeed, since there are no edges for the graph E3, each vertex can be colored with any of the λ colors, and because we have 3 vertices, we get a total of λ3 colorings. When λ = 1 we have 13 = 1 so, there is one way to color the graph with one color option. Moreover, let’s say λ = 2. We would then expect there to be 8 different ways to color the graph using two colors. This can be easily seen by trying to color by hand.

Figure 3.1 E3 graph

Example 3.3.2. Say we have an empty graph En with n vertices and no edges. Since, there are no edges each vertex can be colored in λ ways. Thus, we conclude that the chromatic polynomial is

n P (En, λ) = λ .

Example 3.3.3. Another family of graphs we have discussed is the path graph, Pn. We will start

by finding the chromatic polynomial of P4. We begin by trying to color the graph using 1 color. If

we try to color the graph P4 in Figure 3.2 using one color, we notice that it is not possible using our definition of proper coloring. This means that there are zero ways to color this graph using 21 one color, namely P (P4, 1) = 0. Now, say that we have two colors to choose from. Starting to color the graph in Figure 3.2 from left to right, we have two options for the first vertex, but only one option for the second, third, and fourth vertex. So P (P4, 2) = 2. Figure 3.2 shows a proper coloring using two colors, blue and yellow.

Finally, let’s say we have λ color options. This means that we will have λ ways to color the first vertex because we can choose any color we want. Then, we have to choose a different color for the adjacent vertex, meaning we have λ − 1 options to choose from for the second vertex. Continuing to the third vertex, we have λ−1 options, meaning we can choose any color other than the adjacent vertex. Finally, for the last vertex, we have again λ − 1 options. So the P4 chromatic polynomial

3 is P (P4, λ) = λ(λ − 1) . This polynomial holds true with our previous remarks when λ = 1 or

n−1 λ = 2. We can generalize this example to the case Pn and show that P (Pn, λ) = λ(λ − 1) .

Figure 3.2 Coloring of P4 graph

Example 3.3.4. Next, we will find the chromatic polynomial of a complete graph. We begin with

the graph K5, a complete graph with five vertices. As seen previously, each vertex will be colored a different color since all of the vertices are adjoined by an edge. Referring to Figure 3.3, we can first think about coloring using one color, then two colors, then three colors, and then four colors. Notice that in each case we can not properly color the graph using four or less colors. This means

that P (K5, λ) = 0 if λ ≤ 4. However, we can color the graph using five colors. In fact, we have five possibilities to color the first vertex, four possibilities to color the second vertex, three

for the third vertex, two for the fourth, and finally one option for the last. So, P (K5, λ = 5) = 5(4)(3)(2)(1) = 120. More generally, assuming we have λ colors, where λ ≥ 5. For the first vertex, we have λ possibilities. For the second vertex, we have λ − 1 possibilities. For the third vertex, we have λ − 2 possibilities. Similarly, we have λ − 3, and λ − 4, possibilities for vertices four and five

respectively. Thus, our chromatic polynomial can be written as P (K5, λ) = λ(λ − 1)...(λ − 4). 22

One can easily generalize this to a Kn graph and get that the chromatic polynomial as P (Kn, λ) = λ(λ − 1)...(λ − (n − 1)).

Figure 3.3 Proper coloring of K5 graph

Sometimes we will encounter graphs that we are unfamiliar with and we cannot compute the chromatic polynomial just by using the definition. Thus, we need a strategy that can help us to calculate the chromatic polynomial for these unknown graphs. There are two important operations that we can perform on G that help us to do induction. The first operation is called an edge deletion, whereas, the second operation is called edge contraction. Each of these operations help us to reduce our problem from a complicated graph into a slightly simpler graph. Edge deletion will give us fewer edges, while edge contraction will give us fewer vertices. We can then repeat these operations until we have a graph that we recognize.

3.4 Edge Deletion and Contraction

Definition 3.4.1. Let G be a graph and e an edge of G. The graph G − e is a graph with the same vertices as G, and the same edges, except we dont have the edge e.

Example 3.4.2. Looking at the Figure 3.4, we can see an edge deletion of the edge adjoining vertex 1 and vertex 3. Notice how there are the same amount of vertices for both graphs.

Definition 3.4.3. Let G be a graph and e be an edge of G with vertices x and y; e = (x, y). We define a new graph G/e by taking all the vertices of G that are different from x and y, and replace the two vertices x, y with a single vertex z. For each edge in G that does not use x or y, add it into 23

1 2

a) 4 3

1 2

b) 4 3

Figure 3.4 An example of edge deletion, e = (1, 3)

G/e. For each vertex v different from x and y, we will have an edge between v and z in G/e, if we have an edge between v and x or an edge between v and y in G.

Example 3.4.4. Let G be the graph in Figure 3.5(a). We want to use the operation of edge con- traction for the edge e = (1, 4). As described above, we will replace the two vertices of 1 and 4 with a single vertex, 1 and we get the graph in Figure 3.5(b).

1 2

a) 4 3

1 2

b) 3

Figure 3.5 Contraction of the edge (1,4)

The following theorem is used to compute chromatic polynomial of a graph. It also shows that P (G, λ) is actually a polynomial. 24 Theorem 3.4.5. Take G be a simple graph, and e an edge in G. Let G − e and G/e be the graphs obtained from G by deleting, contracting the edge e, respectively. Then

P (G, λ) = P (G − e, λ) − P (G/e, λ).

Proof. Let e = (v, w) where v, w ∈ V (G). The number of λ-colorings of G − e in which v and w have different colors is the same with or without edge e, and thus equal to P (G, λ). Similarly, the number of λ-colorings of G − e in which v and w have the same color does not change regardless of whether the two vertices are contracted; this number is thus equal to P (G/e, λ). As a result, we find that the total number of λ-colorings of G − e is P (G, λ) + P (G/e, λ). Subtraction yields P (G, λ) = P (G − e, λ) − P (G/e, λ) as desired.

Remark 3.4.6. Theorem 3.4.5 implies that P (G, λ) is a polynomial. Indeed, when we apply The- orem 3.4.5 several times we eventually reduce P (G, λ) to a sum of P (Ek, λ), and we know that

k P k P (Nk, λ) = λ , which implies P (G, λ) = λ ∈ Z[λ] is a polynomial.

This theorem gives us an algorithm to reduce the complexity of a graph into a simpler form. We will now discuss a few examples.

Example 3.4.7. We know from our previous discussion about the Cn graph that the chromatic number must be either 3 or 2, depending on whether Cn is odd or even, respectively. This means that we would expect the C4 graph to have a chromatic number of 2, meaning that the chromatic polynomial evaluated at 2 is a non-zero number. Although, we are aware of the chromatic number, we might not be familiar with the chromatic polynomial.

Consider the graph C4 in Figure 3.6. Take e to be the edge that connects 1 and 4. We can use

Theorem 3.4.5 to reduce the the C4 graph into two graphs that we are familiar with, a P4 and K3 graph. (See Figure 3.7)

Using the fundamental reduction theorem, PG(λ) = PG−e(λ) − PG/e(λ), and the values for 25 1 2

4 3

Figure 3.6 The graph C4

1 2 1 2

4 3 3

Figure 3.7 C4 − e = P4 and C4/e = K3

the chromatic polynomial of the graphs in Figure 3.7, a P4 graph and a K3 graph, to find the chromatic polynomial of the C4 graph in Figure 3.6.

The first graph in Figure 3.7 is a path graph of length 4. The chromatic polynomial of a P4 graph can be calculated as (λ)(λ − 1)3. Shown in Example 3.3.3.

Finally, the second graph in 3.7 is the complete graph with 3 vertices, K3. The K3 graph has the chromatic polynomial of P (K3, λ) = λ(λ − 1)(λ − 2). By calculating the difference between the two chromatic polynomials in Figure 3.7, we can calculate the chromatic polynomial of the C4.

P (C4, λ) = P (C4 − e, λ) − P (C4/e, λ)

= P (P4, λ) − P (K3, λ) = (λ)(λ − 1)3 − λ(λ − 1)(λ − 2) = (λ)(λ − 1)[(λ2 − 2λ + 1) − (λ − 2)] = (λ)(λ − 1)(λ2 − 3λ + 3) = λ4 − 4λ3 + 6λ2 − 3λ = (λ − 1)4 + (λ − 1)

Remark 3.4.8. Furthermore, one can use edge deletion and contraction to compute chromatic polynomial for Cn. In general, we can show that: 26 n n P (Cn, λ) = (λ − 1) + (−1) (λ − 1).

Example 3.4.9. Now we will look at a slightly more complicated example, starting with a graph G that we are not familiar with shown in Figure 3.8. Take the edge e = (2, 4). The Figure 3.9 shows

that G − e and G/e are two graphs that we are familiar with, a C4 graph and a P3 graph.

1 2

4 3

Figure 3.8 A graph G, e = (2, 4)

1 2 1 2

4 3 3

Figure 3.9 G − e = C4 and G/e = P3

We know from Example 3.4.7 that the chromatic polynomial of a C4 graph can be expressed

4 4 as (λ − 1) + (−1) (λ − 1), and that the chromatic number of a P3 graph can be calculated as (λ)(λ − 1)2. By calculating the difference of the two chromatic polynomials, we can get

PG(λ) = PG−e(λ) − PG/e(λ) = (λ − 1)4 + (−1)4(λ − 1) − (λ)(λ − 1)2

Remark 3.4.10. Notice that P (C4, 2) = 2, so the graph C4 can be colored with 2 colors. Also, P (G, 2) = 14 + (−1)4 − 2(1)2 = 0 and so G cannot be colored with 2 colors. However P (G, 3) = 24 + 2 − 3 · 4 6= 0. So, G can be colored with 3 colors. Which shows that χ(G) = 3. 27

CHAPTER 4 SPLIT VERTICES

4.1 Scheduling

As mentioned in Chapter one, we can use graphs to model a wide array of natural phenomena. Using proper coloring, we can find solutions to many real world questions. For example, one way we can use graphs is to improve scheduling. To address this, let’s look at a basic example.

Example 4.1.1. Suppose that an assortment of committees needs to be scheduled, each for a total of one hour. Certain pairs of committees, because they have a member in common, cannot meet at the same time. What is the length of the shortest time interval in which all the committees can be scheduled?

Let G be the graph whose vertices are these committees, with an edge between two committees if they cannot meet at the same time. The standard answer to the scheduling question is that the length of the shortest time interval is the chromatic number of G. To be more concrete, suppose there are five committees (A, B, C, D, E), with scheduling con- flicts. Committee A and B have a member in common, committee B and C have a member in common, committee C and D have a member in common, committee D and E have a member in common, and finally committee E and A have a member in common. We can set up this problem using our intuition. If we let committee A and C meet in the first hour, B and D meet in the second hour, and finally E meet in the 3rd hour, we can create a schedule for all of the committees to meet in 3 hours total. Let’s assume that the first meeting begins at 1:00pm so, the final meeting would end at 4:00pm. We get the schedule in Table 4.1.

Time Committee A Committee B Committee C Commitee D Committee E 1:00pm-2:00pm X X 2:00pm-3:00pm X X 3:00pm-4:00pm X

Table 4.1 Schedule 1 28 Clearly, we can create a schedule for the 5 committees in 3 hours. We can also use graphs to represent this information to prove that 3 hours is the smallest amount of time that the committees can all have their meetings given that they each meet for a full hour. Each committee we can represent using a vertex. For each of the committees that can not meet at the same time, we will adjoin the vertices with an edge. For the example discussed above we get the graph C5 to represent the scenario as shown in Figure 4.1. Each color that we use to color the graph represents a time slot that the committees meet.

A E

B

D C

Figure 4.1 Committees Graph Coloring

The graph C5, can be colored with three colors, meaning that the scheduling can be done in three hours. From the previous chapter, we know that the chromatic polynomial for the cycle

5 graph, C5 is P (C5, λ) = −(λ − 1) + (λ − 1) . So letting λ = 2, we get P (C5, λ = 2)=0. Thus, we cannot color C5 with 2 colors. Although this graphical representation is helpful in finding the minimum number of colors, we are limited by thinking that the schedule cannot be improved. In fact, the schedule can be improved by half an hour if the committees can be scheduled in half hour time blocks rather than one hour time blocks. As you can see from Table 4.2, each committee is still meeting for a full hour in total, but we managed to create a schedule in which all of the committees meet from 1:00-3:30pm, saving a half hour of time compared to the first schedule. Thus, we can improve the schedule to be meeting in 2.5 hours rather than 3 hours. 29 Time Committee A Committee B Committee C Commitee D Committee E 1:00pm-1:30pm X X 1:30pm-2:00pm X X 2:00pm-2:30pm X X 2:30pm-3:00pm X X 3:00pm-3:30pm X X

Table 4.2 Schedule 2

What does this tell us about our graph? Currently, looking at Figure 4.1, we have designed each of the vertices to represent one full hour. We can take that same graph and split the vertices to represent half hours. This shows us that when we allow the vertices to be split, we can produce a more efficient graph so the chromatic number of the graph with split vertices should be lower than without split vertices. We can represent the half hour time increments by each half of the vertex. Notice that we again adjoin the committees that cannot meet at the same time with an edge. To color our graph, we need to define split proper coloring.

4.2 Definitions

In order to translate the ideas from the previous section to graph language we need some def- initions. Our goal is to color each vertex with two colors so that the vertices adjoined with an edge have different colors. To define split proper coloring, we will modify our definition of proper coloring.

Definition 4.2.1. Let G be a graph and C = {c1, c2, ..., cλ} a set of colors. A proper λ-split coloring of G is a function f : V → C × C so that f(v) = (c1, c2) with, c1 6= c2, and ci ∈ C. More over, if e is an edge from v to u such that f(v) = (c1, c2), and f(u) = (d1, d2) then ci 6= dj for all 1 ≤ i, j ≤ 2.

Definition 4.2.2. The minimum number of colors needed to proper split color a graph G will be defined as the split chromatic number of G and denoted by sχ(G).

The definition of the split chromatic polynomial using the new definition of proper split color- ing is as follows. 30 Definition 4.2.3. Let G be a graph. The split chromatic polynomial sP (G, x) ∈ Z[x] of G is defined by sP (G, λ) = the number of distinct proper split λ-colorings of G for every λ ∈ N.

Using these definitions, we can discuss how to split color the C5 graph from Figure 4.1. Using the information from Table 4.2, we can split color the graph in Figure 4.2 as follows. Committee A will be meeting during the times 1:00-1:30pm, represented with the color red, and will be meeting 1:30-2:00pm represented with the color orange. By following the schedule we can color in the remaining vertices of the graph with 2:00pm-2:30pm represented with the color yellow, 2:30-3:00pm represented with the color blue, and finally 3:00pm to 3:30pm represented with the color green. Since we have used five colors for five split vertices, the time for the five committees to meet would be 5/2 = 2.5 hours.

A E

B

C

D

Figure 4.2 Committees with Split Vertices

4.3 The Split Chromatic Polynomial

This real world example is what motivates us to study the use of splitting vertices and leads us to the question of how this relates to the chromatic polynomials and graphs without split vertices. Start with a graph G = (V,E). We construct a new graph DG as follows:

0 00 0 00 0 00 If V (G) = {v1, v2, v3, ..., vn}. Then V (DG) = {v1, v1 , v2, v2 , ..., vn, vn}. In particular, |V (DG)| = 2|V (G)|

Take an edge e ∈ E(G) that connects v and w then we will have edges e0, e00, e000, e0000 ∈ E(DG) e0 from v0 to w0 31

e f

Figure 4.3 Split Coloring

e00 from v00 to w00 e000 from v0 to w00 e0000 from v00 to w0

0 00 Also, for every vertex v ∈ V (G) we have an edge ev ∈ E(DG) that connects v to v . We

0 00 000 0000 S define E(DG) = {e , e , e , e | e ∈ E(G)} {ev | v ∈ V (G)}. In particular, |E(DG)| = 4|E(G)| + |V (G)|.

Remark 4.3.1. One can easily see that to give a proper coloring of DG is equivalent to give a split coloring for G. In particular, we have that sP (G, λ) = P (DG, λ).

We will look at a simple example to construct the graph DG. We will transform the graph G of Figure 4.3 into a graph DG using the algorithm described above.

Example 4.3.2. Consider the graph P3 in Figure 4.3. First, make each of the halves into two different vertices adjoined with an edge. So, we have that our first vertex becomes vertex 10 and vertex 100, our second split vertex becomes vertex 20 and 200, and our last split vertex becomes vertex 30 and 300. Then each of these vertices should be adjoined to the vertices that previously shared an edge and to its pair vertex. Thus, we have vertex 10, 100, 20, 200 should be connected among

0 00 0 00 themselves. Vertex 2 , 2 , 3 , 3 should be connected among themselves. Which means that DP3 is the graph in Figure 4.4. Notice that it can be more easily seen how to properly color this graph. Additionally, we can find the chromatic polynomial to prove that the chromatic number of this graph is four.

Example 4.3.3. To write the chromatic polynomial of the graph in Figure 4.4, we will start by naming the graph G and taking λ colors. Vertex 10 can be colored in λ ways, vertex 100 in λ − 1 ways, vertex 20 in λ−2 ways, vertex 200 in λ−3 ways, vertex 30 in λ−2 ways, and finally vertex 300 in 32

100 00 300 e00 2 f 00

e1 e2 e3

0 0 0 1 e0 2 f 0 3

Figure 4.4 DP3

λ−3 ways. Thus the chromatic polynomial can be written as P (G, λ) = λ(λ−1)(λ−2)2(λ−3)2. Hence, we need a minimum of four colors to color this graph.

To give a different proof of the fact that the chromatic polynomial of the graph in Figure 4.4 is P (G, λ) = λ(λ − 1)(λ − 2)2(λ − 3)2, we can use edge contraction and deletion to transform the graph into something that we are already familiar with. The goal is to transform the graph into complete graphs and empty graph because we can then calculate the chromatic polynomial. After our first edge deletion and contraction, we see the two graphs shown in Figure 4.6. We can continue this process for many more steps and we end up with twelve graphs. Each of these graphs look like on of the three graphs shown in Figure 4.7. Moreover, we see one copy of the first graph, we subtract three copies of the second graph, and we add six copies of the last graph. The

first graph is a K4 graph with two empty graphs. Thus, the chromatic polynomial is P (G1, λ) =

3 λ (λ − 1)(λ − 2)(λ − 3). The second graph is a K4 graph with one empty graph with a chromatic

2 polynomial P (G2, λ) = λ (λ − 1)(λ − 2)(λ − 3). Finally the last graph shown is the K4 graph

with chromatic polynomial P (G3, λ) = λ(λ − 1)(λ − 2)(λ − 3). When we do the calculation, we see the following:

3 2 sP (P3, λ) = λ (λ − 1)(λ − 2)(λ − 3) − 5λ (λ − 1)(λ − 2)(λ − 3) + 6λ(λ − 1)(λ − 2)(λ − 3) = λ(λ − 1)(λ − 2)(λ − 3)(λ2 − 5λ + 6) = λ(λ − 1)(λ − 2)(λ − 3)(λ − 2)(λ − 3) = λ(λ − 1)(λ − 2)2(λ − 3)2

Thus, we conclude the the split chromatic polynomial of the graph P3 is sP (P3, λ) = λ(λ − 1)(λ − 2)2(λ − 3)2 33 1 3 5

2 4 6

Figure 4.5 G = DP3

1 3 5 1 3 5

2 4 6 2 4

Figure 4.6 The graph of G after one deletion and contraction

1 3 5

2 4 6 G1

1 3 5

2 4 G2

1 3

2 4 G3

Figure 4.7 The 3 types of graphs of G with edge contraction and deletion

We can continue this process with other types of graphs to compute the split chromatic poly- nomial of G by finding the chromatic polynomial of DG.

Example 4.3.4. Next we discuss the empty graph, E3. One can easily see that DE3 is the graph 34

Figure 4.8 Split Coloring of E3

3 3 in Figure 4.9. So sP (E3, λ) = P (DE3, λ) = λ (λ − 1) . More generally, one can show that

n n sP (En, λ) = P (DEn, λ)=λ (λ − 1) .

2 4 6

1 3 5

Figure 4.9 DE3

Example 4.3.5. Consider the graph K5 shown in Figure 4.10. One can easily see that DK5 = K10.

See Figure 4.11. We know that the chromatic polynomial of P (K10, λ) = λ(λ − 1)(λ − 2)(λ −

3)...(λ − 9). Thus, we can conclude that the split chromatic polynomial of sP (K5, λ) = λ(λ − 1)(λ − 2)(λ − 3)...(λ − 9).

Example 4.3.6. More generally, consider the graph Kn. Just like above DKn = K2n so we have that the split chromatic polynomial of the Kn graph is sP (Kn, λ) = P (K2n, λ) = λ(λ − 1)...(λ − 2n + 1).

Figure 4.10 Split Coloring for K5 35

Figure 4.11 DK5 = K10

Example 4.3.7. We can generalize the path graph example from before. We previously showed

2 2 that the P3 split chromatic polynomial is sP (P3, λ) = λ(λ − 1)(λ − 2) (λ − 2) . If we continue

3 3 this and use edge deletion and contraction, we notice that sP (P4, λ) = λ(λ − 1)(λ − 2) (λ − 3) .

Eventually, we can see a pattern and that the path graph Pn(n ≥ 2) has the split chromatic

n−1 n−1 polynomial of sP (Pn, λ) = λ(λ − 1)(λ − 2) (λ − 3) .

Example 4.3.8. The star graph, Sn is an interesting case to calculate. We can start by calculating the split chromatic polynomial of the S3 graph. The split chromatic polynomial of the S3 graph happens to be the same as the split chromatic polynomial of the P3 graph. Thus, sP (S3, λ) = λ(λ − 1)(λ − 2)2(λ − 2)2.

We will be using edge deletion and contraction to find the split chromatic polynomial for S4. After using edge contraction and deletion, I was able to find three familiar graphs to calculate the split chromatic polynomial. The calculations are computed as follows:

3 2 2 2 2 2 2 2 sP (S4, λ) = λ (λ − 1)(λ − 2) (λ − 3) − 5λ (λ − 1)(λ − 2) (λ − 3) + 6λ(λ − 1)(λ − 2) (λ − 3) =λ(λ − 1)(λ − 2)2(λ − 3)2(λ2 − 5λ + 6) =λ(λ − 1)(λ − 2)2(λ − 3)2(λ − 2)(λ − 3)

3 3 so sP (S4, λ) = λ(λ − 1)(λ − 2) (λ − 3)

We can also calculate the chromatic polynomial of S5 using what we found from S4. After many edge deletions and contractions, you are left with 9 familiar graphs that you can write the chromatic polynomial. After my calculations, I found the following: 36 5 2 2 4 2 2 3 2 sP (S5, λ) = λ (λ − 1)(λ − 2) (λ − 3) − 5λ (λ − 1)(λ − 2) (λ − 3) + 6λ (λ − 1)(λ − 2) (λ − 3)2 − 5λ4(λ − 1)(λ − 2)2(λ − 3)2 + 25λ3(λ − 1)(λ − 2)2(λ − 3)2 − 30λ2(λ − 1)(λ − 2)2(λ − 3)2 + 6λ3(λ − 1)(λ − 2)2(λ − 3)2 − 30λ2(λ − 1)(λ − 2)2(λ − 3)2 + 36λ(λ − 1)(λ − 2)2(λ − 3)2 =λ5(λ − 1)(λ − 2)2(λ − 3)2 − 10λ4(λ − 1)(λ − 2)2(λ − 3)2 + 37λ3(λ − 1)(λ − 2)2(λ − 3)2 − 60λ2(λ − 1)(λ − 2)2(λ − 3)2 + 36λ(λ − 1)(λ − 2)2(λ − 3)2 =λ(λ − 1)(λ − 2)2(λ − 3)2(λ4 − 10λ3 + 37λ2 − 60λ + 36) =λ(λ − 1)(λ − 2)2(λ − 3)2(λ − 2)2(λ − 3)2

4 4 so sP (S5, λ) = λ(λ − 1)(λ − 2) (λ − 3)

If we repeat this process again, we will end up with 27 familiar graphs that we can find from transforming the of S6. As you can see, the process becomes a lot more tedious and can easily become a lot more complicated to calculate. Luckily, we can generalize our results. More precisely, we can show that the split chromatic polynomial of Sn is the same as the split chromatic polynomial of Pn

n−1 n−1 sP (Sn, λ) = λ(λ − 1)(λ − 2) (λ − 3)

.

Figure 4.12 Split Coloring of S4 37

Figure 4.13 Split Coloring of S5

8 7

2 4 6

1 3 5

Figure 4.14 DS4

Remark 4.3.9. The cycle graph becomes quite difficult to calculate. Theoretically, it can be cal- culated using the same edge deletion and contraction method, but we were not able to find a nice general formula. 38 8 7

2 4 6

1 3 5

9 10

Figure 4.15 DS5

Graph Type Chromatic Polynomial Split Chromatic Polynomial χ(G) sχ(G) n n En λ (λ(λ − 1)) 1 2 n−1 n−1 n−1 Pn λ(λ − 1) λ(λ−1)(λ−2) (λ−3) 2 4 n−1 n−1 n−1 Sn λ(λ − 1) λ(λ−1)(λ−2) (λ−3) 2 4 Kn λ(λ−1)(λ−2)...(λ−(n−1)) λ(λ − 1)...(λ − (2n − 1)) n 2n n n Cn (−1) (λ − 1) + (λ − 1) ? 2 or 3 4 or 5

Table 4.3 Summary of Chromatic Polynomials of Graphs

To summarize our results, we have Table 4.3.

Remark 4.3.10. When n = 3, the graph C3 = K3 and so, we know the split chromatic polynomial of C3. For n = 4, it is quite tedious to compute the chromatic polynomial of DC4 by hand. Eventually, one could try to generalize this information and compute the chromatic polynomial of

DCn, but we were not able to do that.

Remark 4.3.11. There are no good reasons to restrict ourselves to splitting a vertex in only two parts. One can consider splitting in three or more parts. With the appropriate change, we can talk about fractional chromatic number and fractional chromatic polynomial. These topics were studied over the years. For example see Karafov and Sotk (2015) and Scheinerman and Ullman (2008). Our work from this paper was done independently from these results and references. 39

CHAPTER 5 NEIGHBORING VERTICES

5.1 Introduction

Previously we have discussed how to properly color a graph. By using proper coloring, it was discovered that any map can be colored using four colors. An example of this is shown in Figure 5.1. Notice that Figure 5.1 shows the map of Midwest and bordering states to be colored using the four colors red, yellow, green, and orange. Furthermore, notice that some of the bordering states of Ohio are the same color. For example, Michigan, Pennsylvania, and Kentucky are all the color green.

Figure 5.1 Map of Midwest

To represent Figure 5.1 we can use graph theory. Since the states that share a border can not share the same color, we will have an edge between them. By doing so, we are able to see that the map of Ohio and its neighboring states creates the graph shown in Figure 5.2. Using proper coloring, we can conclude that you need a minimum of three colors to color the graph. Let’s say we want to make the coloring of the map a little more interesting. Rather than having only the states that share borders be different colors; we will add an additional condition so that all 40 WI MI

IL IN OH PA

KY WV

Figure 5.2 Graph of Midwest

of the bordering states of one state must be different colors as well. In our specific case, we want the states that border Ohio to be distinct colors. We will start by creating a graph that represents this information. Previously, we adjoined neighboring states with an edge. We keep this condition, but we will also adjoin the states that share the border of Ohio with an edge. For example, we will connect Kentucky to not only Ohio, Indiana, Illinois and West Virginia, but we will also connect Kentucky with Michigan, Wisconsin, Pennsylvania since we don’t want them to have the same colors. Now after making these new con- nections, we can see that the map of the Midwest States and its neighbors distance two connected with edges creates a new graph, see Figure 5.3, that takes more than four colors to be colored. In fact, if we ignore Wisconsin and Illinois for a moment, we can see that the remaining states create

a complete K6 graph. So we know just by looking at a K6 graph, we need at least 6 colors because the chromatic polynomial is P (K6, λ) = (λ)(λ − 1)(λ − 2)(λ − 3)(λ − 4)(λ − 5). We can color the graph using our intuition as shown in Figure 5.4. Illinois is at a distance of 3 states away from Pennsylvania so, Illinois and Pennsylvania can be colored the same color. In this example, we colored these states blue. Since Wisconsin is more than distance two states away from Pennsylvania, and West Virginia; Wisconsin can have the same color as any of these states. In this case, we used yellow to color Wisconsin and West Virginia. The rest of the graph must be colored different colors since each of the vertices are adjacent to one another. Hence, the minimal amount of colors needed to color this graph using distance two coloring would be 6. To prove that 6 is the minimum number of colors needed, we can use edge deletion and con- 41 WI

MI OH

IL PA

IN WV KY

Figure 5.3 Graph of Midwest with New Condition

Y

R G

B B

O Y P

Figure 5.4 Distance Two Graph Coloring of Midwest

traction to find the chromatic polynomial of the graph in Figure 5.3.

5.2 Definitions

We want to formalize the example described above. This means that we must modify our original definition of proper coloring.

Definition 5.2.1. (Distance two proper λ coloring). Let G be a graph and let λ ∈ N. A proper distance two λ-coloring of a simple graph G = (V,E) is an assignment of colors to the vertices of a graph so that no two adjacent vertices and no vertices connected by a path of length at most two have the same color.

The definition of distance two the chromatic polynomial is based on the above definition. 42 Definition 5.2.2. Let G be a graph and λN. The distance two chromatic polynomial nP (G, λ) of G is defined as the number of distinct proper distance two λ-colorings of G.

Let G = (V,E) be a graph. We create a new graph NG as follows. The vertices for NG are the same as the vertices for G. If e is an edge in G the e is also an edge in NG. Finally, if x, y ∈ V (G) such that there exists z ∈ V (G) with (x, z) and (z, y) edges in G then we add an edge fx,y in NG. Notice the relationship between distance two coloring for G and regular proper coloring for NG. We can see that by coloring G using distance two coloring it is the same as coloring NG using regular proper coloring techniques. In particular, one can easily see that nP (G, λ) = P (NG, λ).

5.3 Examples

Example 5.3.1. Let us discuss the empty graph, E3. One can easily see that NE3 is the same as E3

3 because no vertices are distance two away from one another. So nP (E3, λ) = P (NE3, λ) = λ .

n More generally, one can show that nP (En, λ) = P (NEn, λ) = λ .

Example 5.3.2. Consider the graph K5. One can easily see that N(K5) = K5 since all of the vertices of distance two are already connected. We know that the chromatic polynomial of

P (K5, λ) = λ(λ − 1)(λ − 2)(λ − 3)(λ − 4). Thus, we can conclude that the distance two chro-

matic polynomial nP (K5, λ) = λ(λ−1)(λ−2)(λ−3)(λ−4). More generally, consider the graph

Kn. Just like above N(Kn) = Kn so we have that the distance two chromatic polynomial of the

Kn graph is nP (Kn, λ) = P (Kn, λ) = λ(λ − 1)...(λ − n + 1).

Recall the star graph with six vertices from Figure 5.5. Previously, we could see that the star graph could be colored with a minimum of λ = 2 colors. Additionally, it can be easily shown that

n−1 star graph with n spokes has the chromatic polynomial P (Sn, λ) = λ(λ − 1) . So, for the S6

5 graph as seen in Figure 5.5, we have the chromatic polynomial P (S6, λ) = λ(λ − 1) . 43 5

4 6 1

2 3

Figure 5.5 The S6 graph

So what would happen to the graph if we want none of the distance two neighbors to be the same color? Using our definition, we want connected vertices with an path of at most two to

be different colors. This means that for the S6 graph, none of the vertices can share the same color. As a result, we would get the same chromatic polynomial and chromatic number as an K6 graph. The chromatic polynomial for a K6 graph is nP (S6, λ) = P (NS6, λ) = P (K6, λ) = λ(λ − 1)(λ − 2)(λ − 3)(λ − 4)(λ − 5). Thus, we would need at least six colors to color the graph.

5 6

4 1

3 2

Figure 5.6 NS6 = K6 graph

Example 5.3.3. Let’s say that we have the graph shown in Figure 5.7, and we want none of the vertices at distance two to have the same color. Notice, we can start by coloring vertex 1 blue, then all of the other vertices cannot be colored blue because they are all at most distance two away. If we color vertex 2 red, then we can not have vertex 1, 3, and 4 the same color, but we can have vertex 5 the same color because it is distance 3 away. 44 1 2 B R

5 4 3 R G O

Figure 5.7 Graph Coloring with distance 2 coloring

Example 5.3.4. Taking the graph shown in Figure 5.8(a), we can see that the graph NG is pre- sented in Figure 5.8(b) by connecting the vertices of distance two. Now, we can more readily

compute the chromatic polynomial P (NG, λ). We can color the first vertex in λ ways, vertex two in λ − 1 ways, vertex three in λ − 2 ways, vertex four in λ − 3 ways and finally ver- tex five can also be colored in λ − 3 ways. So the distance two chromatic polynomial of G is nP (G, λ) = P (NG, λ) = λ(λ − 1)(λ − 2)(λ − 3)(λ − 3). So, we know that we would need at least four colors to give a distance two coloring to the graph which was shown in Figure 5.7.

1 2

4 3

a) 5

1 2

4 3

b) 5

Figure 5.8 Graph G and graph NG

Remark 5.3.5. There is a more general notion of L(h, k) coloring. Our definition of distance two coloring is equivalent to an L(1, 1)-coloring (see Calamoneri (2006)). Again our results were 45 obtained independently from these references. 46

CHAPTER 6 CONCLUSIONS

6.1 Main Results

Throughout this paper, we looked at various ways of using chromatic polynomials to classify different families of graphs. Our main results concerned split colorings and distance two colorings for graphs. More precisely, we were able to show that a proper split coloring of the graph G is the same as a proper coloring of the graph DG. This shows us that the split chromatic polynomial of the graph G is the same as the chromatic polynomial of the graph DG. In short, given λ colors, we have

sP (G, λ) = P (DG, λ).

Given this information, we can compute the chromatic polynomial of the graph DG by using edge deletion and contraction to eventually find the split chromatic polynomial of G. Similarly, we showed that a distance two coloring of the graph G is the same as a distance two coloring of the graph NG. Thus, the distance two chromatic polynomial of the graph G is the same as the chromatic polynomial of the graph NG. So with λ colors, we have

nP (G, λ) = P (NG, λ).

This means that we can again use edge deletion and contraction to eventually find the distance two chromatic polynomial of the graph G.

6.2 Related Questions

There are many interesting questions for future research that can be derived from this paper. Some topics for future work include:

1. One can generalize the split coloring to k-split coloring of a graph G. So it is natural to try

to find a graph DkG such that a k-split coloring of G corresponds to a regular coloring of 47

the graph DkG.

2. Similarly, one can generalize distance two coloring of a graph G to distance k coloring of

G. Can we construct a graph NkG, such that a distance k coloring for G is equivalent with a

regular coloring of NkG?

3. We know from Theorem 3.4.5 that P (G, λ) = P (G − e, λ) − P (G/e, λ). Is there a simi- lar relation for the split chromatic polynomial? Same problem for distance two chromatic polynomial.

4. We are familiar with the Greedy Coloring Algorithm, but is there a way we can improve this algorithm? More precisely, is there a strategy that we can use when we choose the ordering of the vertices that is more likely to produce the chromatic number? 48

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