K-Critical Graphs in $$P_5$$-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.