Constructions of [formula omitted]-critical [formula omitted]-free graphs

Abstract

With respect to a hereditary class C of graphs, a k-chromatic graph G∈C is said to be k-critical if every proper subgraph of G belonging to C is k−1 colorable. It is known that there is a finite number of 4-critical P5-free graphs. We construct an infinite set of k-critical P5-free graphs for every k≥5. We also prove that there are exactly eight 5-critical {P5,C5}-free graphs and thirteen 5-vertex-critical {P5,C5}-free graphs.