Constructions of kk-critical P5P5-free graphs

With respect to a hereditary class CC of graphs, a kk-chromatic graph G∈CG∈C is said to be kk-critical   if every proper subgraph of GG belonging to CC is k−1k−1 colorable. It is known that there is a finite number of 4-critical P5P5-free graphs. We construct an infinite set of kk-critical P5P5-free graphs for every k≥5k≥5. We also prove that there are exactly eight 5-critical {P5,C5}{P5,C5}-free graphs and thirteen 5-vertex-critical {P5,C5}{P5,C5}-free graphs.