Vertex-Critical (\(P_5\), banner)-Free Graphs
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.
Qingqiong Cai was partially supported by National Natural Science Foundation of China (No. 11701297) and Open Project Foundation of Intelligent Information Processing Key Laboratory of Shanxi Province (No. CICIP2018005). Shenwei Huang is partially supported by the National Natural Science Foundation of China (11801284). Tao Li is partially supported by the National Natural Science Foundation (61872200) and the National Key Research and Development Program of China (2018YFB1003405, 2016YFC0400709). Yongtang Shi was partially supported by National Natural Science Foundation of China (Nos. 11771221, 11811540390), Natural Science Foundation of Tianjin (No. 17JCQNJC00300), and the China-Slovenia bilateral project “Some topics in modern graph theory” (No. 12-6).
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