Forced color classes, intersection graphs and the strong perfect graph conjecture

Abstract

In 1996, A. Sebő[11] raised the following two conjectures concerned with the famous Strong Perfect Graph Conjecture: (1) Suppose that a minimally imperfect graph G has a vertex p incident to 2ω(G)−2 determined edges and that its complement G ̄ has a vertex q incident to 2α(G)−2 determined edges. ( An edge of G is called determined if an ω- clique of G contains both of its endpoints.) Then G is an odd hole or an odd antihole. (2) Let v 0 be a vertex of a partitionable graph G. And suppose A,B to be ω- cliques of G so that v 0 ∈ A ∩ B . If every ω- clique K containing the vertex v 0 is contained in A ∪ B , then G is an odd hole or an odd antihole. In this paper, we will prove (1) for a minimally imperfect graph G such that (p,q) is a determined edge of either G or G ̄ , and prove (2) for a minimally imperfect graph G such that G ̄ is C 4 -free and edges of G ̄ are all determined edges.