  For $k\geq 1$, a $k$-colouring $c$ of $G$ is a mapping from $V(G)$ to
$\{1,2,\ldots,k\}$ such that $c(u)\neq c(v)$ for any two non-adjacent vertices
$u$ and $v$. The $k$-Colouring problem is to decide if a graph $G$ has a
$k$-colouring. For a family of graphs ${\cal H}$, a graph $G$ is ${\cal
H}$-free if $G$ does not contain any graph from ${\cal H}$ as an induced
subgraph. Let $C_s$ be the $s$-vertex cycle. In previous work (MFCS 2019) we
examined the effect of bounding the diameter on the complexity of $3$-Colouring
for $(C_3,\ldots,C_s)$-free graphs and $H$-free graphs where $H$ is some
polyad. Here, we prove for certain small values of $s$ that $3$-Colouring is
polynomial-time solvable for $C_s$-free graphs of diameter $2$ and
$(C_4,C_s)$-free graphs of diameter $2$. In fact, our results hold for the more
general problem List $3$-Colouring. We complement these results with some
hardness result for diameter $4$.
