Abstract
We study Markov chains for randomly sampling k-colorings of a graph with maximum degree Δ. Our main result is a polynomial upper bound on the mixing time of the single-site update chain known as the Glauber dynamics for planar graphs when k=Ω(Δ/logΔ). Our results can be partially extended to the more general case where the maximum eigenvalue of the adjacency matrix of the graph is at most Δ^(1-ε), for fixed ε > 0.
The main challenge when k≤Δ+1 is the possibility of “frozen” vertices, that is, vertices for which only one color is possible, conditioned on the colors of its neighbors. Indeed, when Δ=Ο(1), even a typical coloring can have a constant fraction of the vertices frozen. Our proofs rely on recent advances in techniques for bounding mixing time using “local uniformity”
Original language | English |
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Pages (from-to) | 731-759 |
Journal | Random Structures & Algorithms |
Volume | 47 |
Issue number | 4 |
Early online date | 1 Jul 2014 |
DOIs | |
Publication status | Published - Dec 2015 |
Keywords
- Markov chain Monte Carlo
- random colorings
- approximate counting
- Glauber dynamics
- mixing time