An Optimal χ-Bound for (P_6, diamond)-Free Graphs
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Given two graphs H_1 and H_2, a graph G is (H_1,H_2)-free if it contains no induced subgraph isomorphic to H_1 or H_2. Let P_t be the path on t vertices and K_t be the complete graph on t vertices. The diamond is the graph obtained from K_4 by removing an edge. In this paper we show that every (P_6, diamond)-free graph G satisfies χ(G)<ω(G)+3, where χ(G) and ω(G) are the chromatic number and clique number of G, respectively. Our bound is attained by the complement of the famous 27-vertex Schläfli graph. Our result unifies previously known results on the existence of linear χ-binding functions for several graph classes. Our proof is based on a reduction via the Strong Perfect Graph Theorem to imperfect (P_6, diamond)-free graphs, a careful analysis of the structure of those graphs, and a computer search that relies on a well-known characterization of 3-colourable (P_6,K_3)-free graphs.
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