Vertex-Critical ($$P_5$$, banner)-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 \(H_2\). Let \(P_t\) and \(C_t\) be the path and the cycle on t vertices, respectively. A banner is the graph obtained from a \(C_4\) by adding a new vertex and making it adjacent to exactly one vertex of the \(C_4\). For a fixed integer \(k\ge 1\), a graph G is said to be k-vertex-critical if the chromatic number of G is k and the removal of any vertex results in a graph with 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 show that there are finitely many 6-vertex-critical (\(P_5\), banner)-free graphs. This is one of the few results on the finiteness of k-vertex-critical graphs when \(k>4\). To prove our result, we use the celebrated Strong Perfect Graph Theorem and well-known properties on k-vertex-critical graphs in a creative way.