On the double-critical graph conjecture

Abstract

A connected graph [Formula: see text] is double-critical if the chromatic number of [Formula: see text] is two less than that of [Formula: see text] whenever [Formula: see text] and [Formula: see text] are two adjacent vertices of [Formula: see text]. The double-critical graph conjecture states that the complete graphs are the only double-critical graphs. We give a proof of this conjecture for any double-critical graph that contains at least one universal vertex. Using this result we prove the conjecture for all double-critical graphs.