Randomly coloring sparse random graphs with fewer colors than the maximum degree
Random Structures & Algorithms
Random Structures & Algorithms
Random sampling of colourings of sparse random graphs with a constant number of colours
Theoretical Computer Science
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 prove that the Glauber dynamics on the C-colorings of a graph G on n vertices with girth g and maximum degree $\Delta$ mixes rapidly if (i) $C=q\Delta$ and $qq^*$, where $q^*=1.4890\ldots$ is the root of $(1-{\rm e}^{-1/q})^2+q{\rm e}^{-1/q}=1$; and (ii) $\Delta\geq D\log n$ and $g\geq D\log\Delta$ for some constant D=D(q). This improves the bound of roughly $1.763\Delta$ obtained by Dyer and Frieze [ Proceedings of the 32nd Annual Symposium on Foundations of Computer Science, 2001] for the same class of graphs. Our bound on this class of graphs is lower than the bound of $11\Delta/6\approx1.833\Delta$ obtained by Vigoda [J. Math. Phys., 41 (2000), pp. 1555--1569] for general graphs.