A Non-Markovian Coupling for Randomly Sampling Colorings

@inproceedings{Hayes2003ANC,
  title={A Non-Markovian Coupling for Randomly Sampling Colorings},
  author={Thomas P. Hayes and Eric Vigoda},
  booktitle={FOCS},
  year={2003}
}
We study a simple Markov chain, known as the Glauber dynamics, for randomly sampling (proper) k-colorings of an input graphG on n vertices with maximum degree ∆ and girth g. We prove the Glauber dynamics is close to the uniform distribution afterO(n log n) steps whenever k > (1 + )∆, for all > 0, assumingg ≥ 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 on the… CONTINUE READING
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