Vertex-Critical (P5, banner)-Free Graphs - Cai, Qingqiong; Huang, Shenwei; Li, Tao; Shi, Yongtang (Frontiers in Algorithmics, 2019)
Given two graphs H1 $$H_1$$ and H2 $$H_2$$ , a graph is (H1,H2) $$(H_1,H_2)$$ -free if it contains no induced subgraph isomorphic to H1 $$H_1$$ or H2 $$H_2$$ . Let Pt $$P_t$$ and Ct $$C_t$$ be the path and the cycle on t vertices, respectively. A banner is the graph obtained from a C4 $$C_4$$ by adding a new vertex and making it adjacent to exactly one vertex of the C4 $$C_4$$ . For a fixed integer k≥1 $$k\ge 1$$ , 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) $$(k-1)$$ -colorable. In this paper, we show that there are finitely many 6-vertex-critical (P5 $$P_5$$ , banner)-free graphs. This is one of the few results on the finiteness of k-vertex-critical graphs when k>4 $$k>4$$ . 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.