Conditions for β-perfectness

A β-perfect graph is a simple graph G such that χ(G′) = β(G′) for every induced subgraph G′ of G, where χ(G′) is the chromatic number of G′, and β(G′) is defined as the maximum over all induced subgraphs H of G′ of the minimum vertex degree in H plus 1 (i.e., δ(H) + 1). The vertices of a β-perfect graph G can be coloured with χ(G) colours in polynomial time (greedily). The main purpose of this paper is to give necessary and sufficient conditions, in terms of forbidden induced subgraphs, for a graph to be β-perfect. We give new sufficient conditions and make improvements to sufficient conditions previously given by others. We also mention a necessary condition which generalizes the fact that no β-perfect graph contains an even hole.