Colouring graphs with no induced six-vertex path or diamond

06/16/2021

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by   Jan Goedgebeur, et al.

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The diamond is the graph obtained by removing an edge from the complete graph on 4 vertices. A graph is (P_6, diamond)-free if it contains no induced subgraph isomorphic to a six-vertex path or a diamond. In this paper we show that the chromatic number of a (P_6, diamond)-free graph G is no larger than the maximum of 6 and the clique number of G. We do this by reducing the problem to imperfect (P_6, diamond)-free graphs via the Strong Perfect Graph Theorem, dividing the imperfect graphs into several cases, and giving a proper colouring for each case. We also show that there is exactly one 6-vertex-critical (P_6, diamond, K_6)-free graph. Together with the Lovász theta function, this gives a polynomial time algorithm to compute the chromatic number of (P_6, diamond)-free graphs.