On Sampling Colorings of Bipartite Graphs

We study the problem of efficiently sampling k-colorings of bipartite graphs. We show that a class of markov chains cannot be used as efficient samplers. Precisely, we show that, for any k, 6 ≤ k ≤ n1/3− , > 0 fixed, almost every bipartite graph on n + n vertices is such that the mixing time of any ergodic markov chain asymptotically uniform on its k-colorings is exponential in n/k (if it is allowed to only change the colors of O(n/k) vertices in a single transition step). This kind of exponential time mixing is called torpid mixing. As a corollary, we show that there are (for every large n) bipartite graphs on 2n vertices with∆(G) = Ω(lnn) such that for every k, 6 ≤ k ≤ ∆/(6 ln∆), each member of a large class of chains mixes torpidly.