MT4514: Graph Theory

Graph Theory 49This completes the inductive step, so the number of edges is at most k 2 for all k.When n is odd the induction is similar, starting with the graph with one vertex,no edges, and no triangles. Definition 1.4. The complement of a graph G has the same vertices as G, butwhenever G has an edge between two vertices the complement has a non-edgebetween those vertices, and whenever G has no edge between two vertices the complementhas an edge.We have seen that it is possible for a graph to have many edges and no triangles.However, in this situation the complement will have many triangles.Example 1.5. Consider the following graph.Here we have 6 vertices, 9 edges and no triangle. The complementary graph (withdashed edges) has two triangles.Ramsey Theory. We proved in Chapter 1, Theorem 2.7 that in a group of sixpeople there are either three who mutually know each other or three who mutuallydon’t know each other. In the previous sentence, we can replace (6, 3) by (18, 4)and it will still be true. These are the prototype Ramsey Theorems. We representpeople by vertices of a complete graph, and colour the edge vw red if v and w knoweach other, and blue otherwise.Note that we are now talking about colouring the edges of a graph, rather thanthe vertices as in Chapter 4.Proposition 1.6. If the edges of K 18 are all coloured either red or blue, then thereis either a red K 4 or a blue K 4 .This proposition will follow from a more general result given in the next section.2. Ramsey NumbersDefinition 2.1. By the Ramsey number R(m, n) we denote the least integer suchthat when the edges of the complete graph K R(m,n) are all coloured either red orblue then the graph contains either a red K m or a blue K n .For example, we have seen that R(3, 3) = 6 and claimed that R(4, 4) = 18.Proposition 2.2. For all m ≥ 2 and n ≥ 2 we haveIn particular, R(m, n) is always finite.R(m, n) ≤ R(m, n − 1) + R(m − 1, n).