On randomly colouring locally sparse graphs

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 for all v ∈ V the average degree of the subgraph Hv induced by the neighbours of v ∈ V is ∆ where ∆ is the maximum degree and ∆ > c1 ln n then for sufficiently large c1, this chain mixes rapidly provided q/∆ > α, where α ≈ 1.763 is the root of α = e. For this class of graphs, which includes planar graphs, triangle free graphs and random graphs Gn,p with p 1, this beats the 11∆/6 bound of Vigoda [20] for general graphs.