The Clique-Rank of 3-Chromatic Perfect Graphs

Abstract

The clique-rank of a perfect graph G introduced by Fonlupt and Sebö is the linear rank of the incidence matrix of the maximum cliques of G. We study this rank for 3-chromatic perfect graphs. We prove that if, in addition, G is diamond-free, G has two distinct colorations. An immediate consequence is that the Strong Perfect Graph Conjecture holds for diamond-free graphs and for graphs with clique number equal to three. The proofs use both linear algebra and combinatorial arguments.