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

AbstractIn 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 Ḡ 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 v0 be a vertex of a partitionable graph G. And suppose A,B to be ω-cliques of G so that v0∈A∩B. If every ω-clique K containing the vertex v0 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 Ḡ, and prove (2) for a minimally imperfect graph G such that Ḡ is C4-free and edges of Ḡ are all determined edges