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For a fixed integer, the $k$-Colouring problem is to decide if the verticesof a graph can be coloured with at most $k$ colours for an integer $k$, suchthat 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$k\geq 3$, 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 followsfrom the known result that the problem is NP-complete even for graphs ofarbitrarily large fixed girth. We examine to what extent the situation maychange if in addition the input graph has bounded diameter.

Colouring Generalized Claw-Free Graphs and Graphs of Large Girth: Bounding the Diameter