A k-colouring of a graph G with colours $1,2,\dots ,k$ is canonical with respect to an ordering $\pi =v_1,v_2,\dots ,v_n$ of the vertices of G if adjacent vertices are assigned different colours and, for $1\le c\le k$ , whenever colour c is assigned to a vertex $v_i$ , each colour less than c has been assigned to a vertex that precedes $v_i$ in $\pi$ . The canonical k-colouring graph of G with respect to π is the graph ${\mathrm{Can}}_k^{\pi }\left(G\right)$ with vertex set equal to the set of canonical k-colourings of G with respect to $\pi$ , with two of these being adjacent if and only if they differ in the colour assigned to exactly one vertex. Connectivity and Hamiltonicity of canonical colouring graphs of bipartite and complete multipartite graphs is studied. It is shown that for complete multipartite graphs, and bipartite graphs there exists a vertex ordering $\pi$ such that ${\mathrm{Can}}_k^{\pi }\left(G\right)$ is connected for large enough values of k. It is proved that a canonical colouring graph of a complete multipartite graph usually does not have a Hamilton cycle, and that there exists a vertex ordering $\pi$ such that ${\mathrm{Can}}_k^{\pi }\left(K_{m,n}\right)$ has a Hamilton path for all $k\ge 3$ . The paper concludes with a detailed consideration of ${\mathrm{Can}}_k^{\pi }\left(K_{2,2,\dots ,2}\right)$ . For each $k\ge \chi$ and all vertex orderings $\pi$ , it is proved that ${\mathrm{Can}}_k^{\pi }\left(K_{2,2,\dots ,2}\right)$ is either disconnected or isomorphic to a particular tree.

The dominating graph of a graph $H$ has as its vertices all dominating sets of $H$, with an edge between two dominating sets if one can be obtained from the other by the addition or deletion of a single vertex of $H$. In this paper we prove that the dominating graph of any tree has a Hamilton path. We also show how a result about binary strings leads to a proof that the dominating graph of a cycle on $n$ vertices has a Hamilton path if and only if $n\not\equiv 0 \pmod 4$.