Randomly colouring graphs with lower bounds on girth and maximum degree

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 Δ>c_l ln n and the girth g>c₂ ln ln n (n=|V|), then this chain mixes rapidly provided C₁, C₂ are sufficiently large, q/A>β, where β≈1.763 is the root of β=e¹β/. For this class of graphs, this beats the 11Δ/6 bound of E. Vigoda (1999) for general graphs. We extend the result to random graphs.