Random generation of combinatorial structures from a uniform
Theoretical Computer Science
Randomized algorithms
A very simple algorithm for estimating the number of k-colorings of a low-degree graph
Random Structures & Algorithms
Simple proofs of occupancy tail bounds
Random Structures & Algorithms
Balls and bins: a study in negative dependence
Random Structures & Algorithms
The Glauber dynamics on colourings of a graph with high girth and maximum degree
STOC '02 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing
Very rapid mixing of the Glauber dynamics for proper colorings on bounded-degree graphs
Random Structures & Algorithms
Path coupling: A technique for proving rapid mixing in Markov chains
FOCS '97 Proceedings of the 38th Annual Symposium on Foundations of Computer Science
Randomly Colouring Graphs with Lower Bounds on Girth and Maximum Degree
FOCS '01 Proceedings of the 42nd IEEE symposium on Foundations of Computer Science
SODA '04 Proceedings of the fifteenth annual ACM-SIAM symposium on Discrete algorithms
Fast mixing for independent sets, colorings and other models on trees
SODA '04 Proceedings of the fifteenth annual ACM-SIAM symposium on Discrete algorithms
Coupling with the stationary distribution and improved sampling for colorings and independent sets
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
Randomly coloring sparse random graphs with fewer colors than the maximum degree
Random Structures & Algorithms
Fast mixing for independent sets, colorings, and other models on trees
Random Structures & Algorithms
Random Structures & Algorithms
Approximate counting via correlation decay in spin systems
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
Randomly colouring graphs with girth five and large maximum degree
LATIN'06 Proceedings of the 7th Latin American conference on Theoretical Informatics
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We improve rapid mixing results for the simple Glauber dynamics designed to generate a random k-coloring of a bounded-degree graph.Let G be a graph with maximum degree Δ = Ω(log n), and girth ≥ 5. We prove that if k Α Δ, where Α ≈ 1.763 then Glauber dynamics has mixing time O(n log n). If girth(G) ≥ 6 and k Β Δ, where Β ≈ 1.489 then Glauber dynamics has mixing time O(n log n). This improves a recent result of Molloy, who proved the same conclusion under the stronger assumptions that Δ=Ω(log n) and girth Ω(log Δ). Our work suggests that rapid mixing results for high girth and degree graphs may extend to general graphs.Analogous results hold for random graphs of average degree up to n¼, compared with polylog(n), which was the best previously known.Some of our proofs rely on a new Chernoff-Hoeffding type bound, which only requires the random variables to be well-behaved with high probability. This tail inequality may be of independent interest.