Some Results on Critical (P₅, H) -free Graphs

Given two graphs H₁ and H₂, a graph is (H₁, H₂) -free if it contains no induced subgraph isomorphic to H₁ nor H₂. A graph G is k-vertex-critical if every proper induced subgraph of G has chromatic number less than k, but G has chromatic number k. The study of k-vertex-critical graphs for specific 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 exists a polynomial-time certifying algorithm to decide the k-colorability of a graph in the class. In this paper, we show that: (1) for k 1, there are finitely many k-vertex-critical (P₅, K₁, ₄+P₁) -free graphs; (2) for s 1, there are finitely many 5-vertex-critical (P₅, K₁, ₒ+P₁) -free graphs; (3) for k 1, there are finitely many k-vertex-critical (P₅, K₃+2P₁) -free graphs. Moreover, we characterize all 5-vertex-critical (P₅, H) -free graphs where H \K₁, ₃+P₁, K₁, ₄+P₁, K₃+2P₁\ using an exhaustive graph generation algorithm.