The structure of critical Ramsey graphs

Abstract

All graphs in this paper are finite and undirected. Let G=(V, E), G'=(V', E'). A one-to-one mapping f : V ~ V " is said to be an embedding iff (x,y)EE,~ ,~,(f(x), f(y))CE'. We say that the graph G' is a Ramsey graph for G if for every partition E'=E1UE2 there exists an embedding f :G~G" such that f ( E ) c E , for an iC {1, 2}. We abbreviate this by G-T+ G'. The negation of this statement is denoted by G ~ G'. Given a graph G, it is an interesting question to characterize all finite graphs G' for which G ~ G'. This seems to be very difficult. Presently it is evident that the structure of all graphs G', which are Ramsey graphs for G, is very complex. In this paper we shall go in this direction using the notion of critical Ramsey graph.