Randomly colouring graphs with lower bounds on girth and maximum degree

Abstract

We consider the problem of generating a random q-colouring of a graph G=(V, E). We consider the simple Glauber Dynamics chain. We show that if the maximum degree /spl Delta/>c/sub l/ ln n and the girth g>c/sub 2/ ln ln n (n=|V|), then this chain mixes rapidly provided C/sub 1/, C/sub 2/ are sufficiently large, q/A>/spl beta/, where /spl beta//spl ap/1.763 is the root of /spl beta/=e/sup 1//spl beta//. For this class of graphs, this beats the 11/spl Delta//6 bound of E. Vigoda (1999) for general graphs. We extend the result to random graphs.