Randomly coloring constant degree graphs

We study a simple Markov chain, known as the Glauber dynamics, for generating a random k -coloring of an n -vertex graph with maximum degree Δ. We prove that, for every ε > 0, the dynamics converges to a random coloring within O(nlog n) steps assuming k ≥ k₀(ε) and either: (i) k/Δ > α* + ε where α*≈ 1.763 and the girth g ≥ 5, or (ii) k/Δ >β * + ε where β*≈ 1.489 and the girth g ≥ 7. Our work improves upon, and builds on, previous results which have similar restrictions on k/Δ and the minimum girth but also required Δ = Ω (log n). The best known result for general graphs is O(nlog n) mixing time when k/Δ > 2 and O(n²) mixing time when k/Δ > 11/6. Related results of Goldberg et al apply when k/Δ > α* for all Δ ≥ 3 on triangle-free “neighborhood-amenable” graphs