  We examine the effect of bounding the diameter for well-studied variants of
the Colouring problem. A colouring is acyclic, star, or injective if any two
colour classes induce a forest, star forest or disjoint union of vertices and
edges, respectively. The corresponding decision problems are Acyclic Colouring,
Star Colouring and Injective Colouring. The last problem is also known as
$L(1,1)$-Labelling and we also consider the framework of $L(a,b)$-Labelling. We
prove a number of (almost-)complete complexity classifications. In particular,
we show that for graphs of diameter at most $d$, Acyclic $3$-Colouring is
polynomial-time solvable if $d\leq 2$ but NP-complete if $d\geq 5$, and Star
$3$-Colouring is polynomial-time solvable if $d\leq 3$ but NP-complete for
$d\geq 8$. As far as we are aware, Star $3$-Colouring is the first problem that
exhibits a complexity jump for some $d\geq 3$. Our third main result is that
$L(1,2)$-Labelling is NP-complete for graphs of diameter $2$; we relate the
latter problem to a special case of Hamiltonian Path.
