On a conjecture about uniquely colorable perfect graphs

AbstractIn a graph G with maximum clique size ω, a clique-pair means two cliques of size ω whose intersection is ω − 1. The subject of this paper is the so-called clique-pair conjecture (CPC) which states that if a uniquely colorable perfect graph is not a clique then it contains a clique-pair. We study the structure of the possible counterexamples to this conjecture, and, combining our results with those in Fonlupt and Zemirline (1987), we obtain a new proof of the CPC for 3-chromatic graphs. Theorem 2 states the validity of the CPC for claw-free graphs