Colouring generalized claw-free graphs and graphs of large girth: Bounding the diameter

For a fixed integer, the k-COLOURING problem is to decide if the vertices of a graph can be coloured with at most k colours for an integer k, such that no two adjacent vertices are coloured alike. A graph G is H-free if G does not contain H as an induced subgraph. It is known that for all k≥3, the k-COLOURING problem is NP-complete for H-free graphs if H contains an induced claw or cycle. The case where H contains a cycle follows from the known result that the problem is NP-complete even for graphs of arbitrarily large fixed girth. We examine to what extent the situation may change if in addition the input graph has bounded diameter.