Vertex-critical [formula omitted]-free graphs

Abstract

Given two graphs H1 and H2, a graph G is (H1,H2)-free if it contains no induced subgraph isomorphic to H1 or H2. A Pt is the path on t vertices. A chair is a P4 with an additional vertex adjacent to one of the middle vertices of the P4. 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. In this paper, we prove that there are only finitely many 5-vertex-critical (P5,chair)-free graphs.