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Randomly coloring constant degree graphs

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  • A preliminary version of this paper appeared in Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science.

Abstract

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 kk0(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.© 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2013

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