Torpid mixing of the Wang-Swendsen-Kotecký algorithm for sampling colorings

We study the problem of sampling uniformly at random from the set of k-colorings of a graph with maximum degree Δ. We focus attention on the Markov chain Monte Carlo method, particularly on a popular Markov chain for this problem, the Wang-Swendsen-Kotecký (WSK) algorithm. The second author recently proved that the WSK algorithm quickly converges to the desired distribution when k⩾11Δ/6. We study how far these positive results can be extended in general. In this note we prove the first non-trivial results on when the WSK algorithm takes exponentially long to reach the stationary distribution and is thus called torpidly mixing. In particular, we show that the WSK algorithm is torpidly mixing on a family of bipartite graphs when 3⩽k<Δ/(20logΔ), and on a family of planar graphs for any number of colors. We also give a family of graphs for which, despite their small chromatic number, the WSK algorithm is not ergodic when k⩽Δ/2, provided k is larger than some absolute constant k0.