Colouring Graphs with No Induced Six-Vertex Path or Diamond

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 Lovasz theta function, this gives a polynomial time algorithm to compute the chromatic number of (\(P_6\), diamond)-free graphs.