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Abstract or Description

We consider the problem of generating a coloring of the random graph n,p uniformly at random using a natural Markov chain algorithm: the Glauber dynamics. We assume that there are βΔ colors available, where Δ is the maximum degree of the graph, and we wish to determine the least β = β(p) such that the distribution is close to uniform in O(n log n) steps of the chain. This problem has been previously studied for n,p in cases where np is relatively small. Here we consider the “dense” cases, where np ε [ω ln n, n] and ω = ω(n) ∞. Our methods are closely tailored to the random graph setting, but we obtain considerably better bounds on β(p) than can be achieved using more general techniques.



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Published In

Random Structures and Algorithms, 36, 3, 251-272.