Minimal vertex Ramsey graphs and minimal forbidden subgraphs

Let P be a property of graphs. A graph G is vertex (P,k)-colourable if the vertex set V(G) of G can be partitioned into k sets V-1, V-2,...,V-k such that the subgraph G[V-i] of G belongs to P, i = 1,2,...,k. If P is a hereditary property, then the set of minimal forbidden subgraphs of P is defined as follows: F(P) = {G:G is not an element of P but each proper subgraph H of G belongs to P}. In this paper we investigate the property O-n : each component of G has at most n+1 vertices. We construct minimal forbidden subgraphs for the property (O-n(k)) "to be (O-n,k)-colourable". We write G-->(v) (H)(k), kgreater than or equal to2, if for each k-colouring V-1, V-2,...,V-k of a graph G there exists i, 1less than or equal toiless than or equal tok, such that the graph induced by the set V-i contains H as a subgraph. A graph G is called (H)(k)-vertex Ramsey minimal if G-->(v) (H)(k), but Gnegated right arrow(v) (H)(k) for any proper subgraph G' of G. The class of (P-3)(k)-vertex Ramsey minimal graphs is investigated. (C) 2003 Elsevier B.V. All rights reserved.