On color-critical ([formula omitted],co-[formula omitted])-free graphs

Abstract

A graph is k-critical if it is k-chromatic but each of its proper induced subgraphs is (k−1)-colorable. It is known that the number of 4-critical P5-free graphs is finite, but there is an infinite number of k-critical P5-free graphs for each k≥5. We show that the number of k-critical (P5,P¯5)-free graphs is finite for every fixed k. Our result implies the existence of a certifying algorithm for k-coloring (P5,P¯5)-free graphs.