  The reconfiguration graph $R_k(G)$ of the $k$-colourings of a graph $G$
contains as its vertex set the $k$-colourings of $G$ and two colourings are
joined by an edge if they differ in colour on just one vertex of $G$.
  We show that for each $k \geq 3$ there is a $k$-colourable weakly chordal
graph $G$ such that $R_{k+1}(G)$ is disconnected. We also introduce a subclass
of $k$-colourable weakly chordal graphs which we call $k$-colourable compact
graphs and show that for each $k$-colourable compact graph $G$ on $n$ vertices,
$R_{k+1}(G)$ has diameter $O(n^2)$. We show that this class contains all
$k$-colourable co-chordal graphs and when $k = 3$ all $3$-colourable $(P_5,
\overline{P_5}, C_5)$-free graphs. We also mention some open problems.
