k-Critical graphs in P5-free graphs

Abstract Given two graphs H 1 and H 2 , a graph G is ( H 1 , H 2 ) -free if it contains no induced subgraph isomorphic to H 1 or H 2 . Let P t be the path on t vertices. A graph G is k-vertex-critical if G has chromatic number k but every proper induced subgraph of G has chromatic number less than k. The study of k-vertex-critical graphs for graph classes is an important topic in algorithmic graph theory because if the number of such graphs that are in a given hereditary graph class is finite, then there is a polynomial-time algorithm to decide if a graph in the class is ( k − 1 ) -colorable. In this paper, we initiate a systematic study of the finiteness of k-vertex-critical graphs in subclasses of P 5 -free graphs. Our main result is a complete classification of the finiteness of k-vertex-critical graphs in the class of ( P 5 , H ) -free graphs for all graphs H on 4 vertices. To obtain the complete dichotomy, we prove the finiteness for four new graphs H using various techniques – such as Ramsey-type arguments and the dual of Dilworth's Theorem – that may be of independent interest.