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We study a simple Markov chain, known as the Glauber dynamics, for randomly sampling (proper) k-colorings of an input graph G on n vertices with maximum degree ∆ and girth g. We prove the Glauber dynamics is close to the uniform distribution after O(n log n) steps whenever k > (1 + )∆, for all > 0, assuming g ≥ 9 and ∆ = Ω(log n). The best previously known bounds were k > 11∆/6 for general graphs, and k > 1.489∆ for graphs satisfying girth and maximum degree requirements. Our proof relies ondoi:10.1109/sfcs.2003.1238234 dblp:conf/focs/HayesV03 fatcat:dff5ua533vdqpopty3sxstakrq