A very simple algorithm for estimating the number of k-colorings of a low-degree graph
Random Structures & Algorithms
Markov Chain Algorithms for Planar Lattice Structures
SIAM Journal on Computing
Randomly Coloring Constant Degree Graphs
FOCS '04 Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science
Mixing in time and space for discrete spin systems
Mixing in time and space for discrete spin systems
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
Random sampling of 3-colorings in ℤ2
Random Structures & Algorithms - Isaac Newton Institute Programme “Computation, Combinatorics and Probability”: Part I
Mixing in time and space for lattice spin systems: A combinatorial view
Random Structures & Algorithms - Isaac Newton Institute Programme “Computation, Combinatorics and Probability”: Part II
A general lower bound for mixing of single-site dynamics on graphs
FOCS '05 Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science
A simple condition implying rapid mixing of single-site dynamics on spin systems
FOCS '06 Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science
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
Randomly coloring random graphs
Random Structures & Algorithms
WINE '08 Proceedings of the 4th International Workshop on Internet and Network Economics
The Glauber Dynamics for Colourings of Bounded Degree Trees
APPROX '09 / RANDOM '09 Proceedings of the 12th International Workshop and 13th International Workshop on Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques
Spectral radius of finite and infinite planar graphs and of graphs of bounded genus
Journal of Combinatorial Theory Series B
Phase transition for the mixing time of the Glauber dynamics for coloring regular trees
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
Using markov-chain mixing time estimates for the analysis of ant colony optimization
Proceedings of the 11th workshop proceedings on Foundations of genetic algorithms
Approximate counting via correlation decay in spin systems
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
Phase transition for Glauber dynamics for independent sets on regular trees
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
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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 knownas 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 graphis at most δ1-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 coloris possible, conditioned on the colors of its neighbors. Indeed, when δ = O(1), even a typical coloring canhave a constant fraction of the vertices frozen.Our proofs rely on recent advances in techniquesfor bounding mixing time using "local uniformity" properties.