Randomly coloring planar graphs with fewer colors than the maximum degree

We study Markov chains for randomly sampling $k$-colorings of a graph with maximum degree $Δ$. Our main result is a polynomial upper bound on the mixing time of the single-site update chain known as the Glauber dynamics for planar graphs when $k=Ω(Δ/\logΔ)$. Our results can be partially extended to the more general case where the maximum eigenvalue of the adjacency matrix of the graph is at most $Δ^{1-\eps}$, for fixed $\eps > 0$. The main challenge when $k \le Δ+ 1$ is the possibility of "frozen" vertices, that is, vertices for which only one color is possible, conditioned on the colors of its neighbors. Indeed, when $Δ= O(1)$, even a typical coloring can have a constant fraction of the vertices frozen. Our proofs rely on recent advances in techniques for bounding mixing time using "local uniformity" properties.

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