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,s+P₁)-free graphs; (3) for k ≥ 1, there are finitely many k-vertex-critical (P₅,K₃+2P₁̄)-free graphs. Moreover, we characterize all $5$-vertex-critical (P₅,H)-free graphs where H ∈ 1,3+P₁,K1,4+P₁,K₃+2P₁̄\ using an exhaustive graph generation algorithm.