A note on graphs of k-colourings

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.