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

Abstract

Given two graphs $H_1$ and $H_2$, a graph is $(H_1,H_2)$-free if it contains no induced subgraph isomorphic to $H_1$ or $H_2$. 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 $k\ge 1$, there are finitely many k-vertex-critical $(P_5,K_{1,4}+P_1)$-free graphs; (2) for $s\ge 1$, there are finitely many 5-vertex-critical $(P_5,K_{1,s}+P_1)$-free graphs; (3) for $k\ge 1$, there are finitely many k-vertex-critical $(P_5,\overline{K_3+2P_1})$-free graphs. Moreover, we characterize all 5-vertex-critical $(P_5,H)$-free graphs where $H\in \{K_{1,3}+P_1,K_{1,4}+P_1,\overline{K_3+2P_1}\}$ using an exhaustive graph generation algorithm.