A note on graphs of $k$-colourings

Abstract

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 $G$.

Figures (3)

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Theorems & Definitions (22)

• Theorem 1
• Conjecture 2: Conjecture 6.1 AKLR24
• Conjecture 3: Conjecture 6.2 AKLR24
• Theorem 4
• Theorem 5
• Theorem 5
• proof
• Lemma 6
• proof
• Lemma 7