A more rapidly mixing Markov chain for graph colorings

We deene a new Markov chain on (proper) k-colourings of graphs, and relate its convergence properties to the maximum degree of the graph. The chain is shown to have bounds on convergence time appreciably better than those for the well-known Jerrum/Salas{Sokal chain in most circumstances. For the case k = 22, we provide a dramatic decrease in running time. We also show improvements whenever the graph is regular, or fewer than 33 colours are used. The results are established using the method of path coupling. We indicate that our analysis is tight by showing that the couplings used are optimal in a sense which we deene.