  For a graph $G$, the $k$-colouring graph of $G$ has vertices corresponding to
proper $k$-colourings of $G$ and edges between colourings that differ at a
single vertex. The graph supports the Glauber dynamics Markov chain for
$k$-colourings, and has been extensively studied from both extremal and
probabilistic perspectives. In this note, we show that for every graph $G$,
there exists $k$ such that $G$ is uniquely determined by its $k$-colouring
graph, confirming two conjectures of Asgarli, Krehbiel, Levinson and Russell.
We further show that no finite family of generalised chromatic polynomials for
$G$, which encode induced subgraph counts of its colouring graphs, uniquely
determine $G$.
