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We study a simple Markov chain, known as the Glauber dynamics, for generating a random k-coloring of a n-vertex graph with maximum degree Δ. We prove that the dynamics converges to a random coloring after O(n log n) steps assuming k ≥ k0 for some absolute constant k0, and either: (i) k/Δ > α* ≈ 1.763 and the girth g ≥ 5, or (ii) k/Δ > β* ≈ 1.489 and the girth g ≥ 6. Previous results on this problem applied when k = Ω(log n), or when k > 11 Δ/6 for general graphs.
Date of Conference: 17-19 Oct. 2004