  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
$k\geq 3$, 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 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.
