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-ε, for fixed ε>0. The main challenge when k≤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.