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 \Delta and girth g. We prove the Glauber dynamics is close to the uniform distribution after 0(n log n) steps whenever k > (1 + \varepsilon)\Delta for all \varepsilon > 0, assuming g ≥ 9 and \Delta = \Omega (\log n). The best previously known bounds were k > 11\Delta/6 for general graphs, and k > 1.489\Delta for graphs satisfying girth and maximum degree requirements.

Our proof relies on the construction and analysis of a non-Markovian coupling. This appears to be the first application of a non-Markovian coupling to substantially improve upon known results.