On Markov Chains for Randomly H-Coloring a Graph

Let H=(W,F) be a graph without multiple edges, but with the possibility of having loops. Let G=(V,E) be a simple graph. A homomorphism c is a map c:V?W with the property that (v,w)?E implies that (c(v),c(w))?F. We will often refer to c(v) as the color of v and c as an H-coloring of G. We consider the problem of choosing a random H-coloring of G by Markov chain Monte Carlo. The probabilistic model we consider includes random proper colorings, random independent sets, and the Widom?Rowlinson and Beach models of statistical physics. We prove negative results for uniform sampling and a positive result for weighted sampling when H is a tree.

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