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