Minimal vertex Ramsey graphs and minimal forbidden subgraphs

Abstract

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∉ 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 , k⩾2, if for each k-colouring V 1, V 2,…, V k of a graph G there exists i, 1⩽ i⩽ k, 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 G′ ↛ v (H) k for any proper subgraph G′ of G. The class of ( P 3) k -vertex Ramsey minimal graphs is investigated.