PRIMES Circle: Graph Theory Tal Berdichevsky and Corinne Mulvey May 17, 2019 Contents 1 Introduction to Graphs 2 1.1 Basics . .2 1.2 Common Types of Graphs . .3 2 Planar Graphs 6 3 Graph Colorings 8 3.1 Introduction to Vertex Coloring . .8 3.2 Four and Five Color Theorems . 10 1 1 Introduction to Graphs How can we model Facebook friends? Here's a diagram of some friends. We can let each point represent an individual person and each line drawn between two points represent a friendship between two people. Abby Bob Clara Dan Eva Fiona Gal Figure 1: Graph of Facebook Friends 1.1 Basics Let's formalize what we just said. The representation we created is an example of a graph. Each friend, represented by the points, can be thought of as a vertex of the graph and the friendships, represented by the lines drawn between points, are edges of the graph. More specifically, Definition 1.1. A graph G is a set of vertices V (G) and edges E(G). Each edge is a \connection" between two vertices, and can be represented as a pair of vertices (u; v). Each edge is said to be incident to the two vertices it connects. Example 1.1. In Figure 1, the vertices are people and edges represent friendships. Abby, Bob, Clara, Dan, Eva, Fiona, and Gal are all vertices and make up the vertex set of the graph. Definition 1.2. Two edges u and v are adjacent if there is an edge between them; in other words, (u; v) 2 E(G). Example 1.2. In Figure 1, any two adjacent vertices, or people, are friends. Definition 1.3. The order and size of a graph G is the cardinality of its vertex set jV (G)j and the cardinality of its edge set jE(G)j, respectively. Example 1.3. The size of the graph in Figure 1 representing Facebook friends is the number of friendships in the network or edges in the graph. The order of graph G is the number of friends in the group. The order and the size of the graph are therefore both equal to 7. Definition 1.4. For a graph G and vertex v 2 V (G), the degree degG(v) of v is the number of edges incident with v. We denote the minimum degree of all vertices in G as δ(G), and the maximum degree we denote as ∆(G). Example 1.4. In Figure 1, a person's degree is the number of friends (s)he has. Clara has two friends and therefore degG(Clara) = 2. The numbers δ(G) and ∆(G) of the social network in Figure 1 represent the people with the least and most friends, respectively. Dan has five friends which is the most friends of anyone in the social network, therefore ∆(G) = 5. Fiona, Gal, and Eva, all each have only one friend, which is the least number of friends anyone in the network has. Therefore δ(G) = 1. Theorem 1.1. For any graph G of size m, X degG(v) = 2m: v2V (G) Proof. When we count the degrees of each vertex in the set V (G), we account for each edge twice, once for each of its two incident vertices. 2 1.2 Common Types of Graphs What happens if Bob and Clara are enemies? Fakebook is an additional social network much like Facebook except that it connects its users if they are enemies instead of friends. As shown in Figure 1.2, we can represent this Fakebook with a graph similarly to how we represented Facebook. We make each person a vertex of the graph and draw an edge between them if they are enemies. Bob Clara Dan Eva Fiona Gal Abby Figure 2: Graph of Fakebook Enemies Say that you wanted to have a party and want to only invite people who do not hate one another. You would need to divide the people into groups so that no \hatred connections" exist in each group. In other words, you would need sets of people who are not adjacent in the Figure 2. Is this possible? As it turns out, this is possible. Dividing the Fakebook enemies into two sets where Fiona, Gal, and Bob are in one set and Clara, Eva, Dan, and Abby are in the other yields a partition where no one hates another person in their respective set. Asking whether these sets exists is the equivalent of asking whether the Fakebook graph is bipartite. Definition 1.5. A bipartite graph is a graph G whose vertex set V (G) can be partitioned into two subsets U and W where no edges of G exist between any two vertices in U or two vertices in W . Definition 1.6. The partite sets of a graph G are the subsets of V (G) in which no vertex is adjacent to another vertex in its respective subset. Example 1.5. The partite sets in the Fakebook example can be described as the sets of people who don't hate one another. As mentioned before, Fiona, Gal, and Bob would make up one partite set and Clara, Eva, Dan, and Abby would make up the other. Therefore, we can say the graph of these Fakebook enemies is a bipartite graph. The graph can be redrawn so these two sets can be seen more clearly. The two partite sets of people who do not hate one another are circled. Notice that in Figure 3 every edge of the graph connects a vertex of one partite set to the vertex of the other partite set. The idea of a bipartite graph can be extended to include graphs which have multiple partite sets or multiple sets of people who don't hate one another. Definition 1.7. A k-partite graph (or a k-colorable graph) is a graph with k partite sets. If every person in each partite set hated every person from the other partite set, then we would have another special type of partite graph. Definition 1.8. A complete bipartite graph is a graph G where every vertex of G is adjacent to every vertex of the other partite set. These graphs are denoted by KU;W with U and W being the cardinality of each partite set. We could also imagine a situation where each person hates everyone else. A graph of this situation on Fakebook would be a graph in which every two vertices (people) are adjacent. In other words, each person hates everyone else in the group. 3 Eva Gal Dan Bob Clara Fiona Abby Figure 3: The Bipartite Graph of Fakebook Enemies Definition 1.9. A complete graph is a graph G where every vertex v 2 V (G) is adjacent to every other vertex u 2 V (G). A complete graph is denoted by Kn where n is the order of graph. Example 1.6. Here is the complete graph of order 4, K4: Definition 1.10. A path Pn is a sequence of n vertices where every two consecutive vertices have an edge between them. Example 1.7. The path P = a; b; c; d; e with E(P ) = ab; bc; cd; de as shown. b d a c e Every vertex in path P has a degree of two except for the first and last vertices a and e which only have a degree of one. If an edge ae was added to P making every vertex have a degree of two, then the resulting graph would no longer be a path. Definition 1.11. A cycle is a graph G similar to a path except the first and last vertices of the path form an edge in G. A cycle of order n is denoted by Cn and can be referred to as a n-gon. Example 1.8. The cycle of order five C5 or the 5-gon, which is simply a pentagon. We might also want to have a new graph H that was a \piece" of another graph G, consisting of some of the vertices and edges of G. 4 Definition 1.12. A graph H = (V 0;E0) is a subgraph of the graph G = (V; E) if V 0 ⊆ V , E0 ⊆ E, and every edge e 2 E0 has its endpoints in V 0. If a subgraph G0 of a graph G is just the graph G with one edge e 2 E(G) or vertex v 2 V (G) missing, then G0 can be denoted as G − e or G − v, respectively. Example 1.9. For example, the path P from before could have a subgraph P 0 with vertex set V (P 0) = a; b; c and edge set E(P 0) = ab; bc . The edge set of P 0 is a subset of P 's edge set and the vertex set of P 0 is a subset of P 's vertex set. Example 1.10. Subgraphs can be formed from some or all of the vertices of a graph. The subgraph H0 of H has the same vertex set as H but only some of the edges of H. H: H0: In most of the graphs we have described, every vertex is \connected" to every other vertex of the graph through some sequence of vertices of edges. Sometimes this connection is just an edge, as when two vertices are adjacent, and sometimes it is a series of adjacent vertices. In a connected graph, there is a path between every two vertices.