Strong spatial mixing for list coloring of graphs

Abstract

The property of spatial mixing and strong spatial mixing in spin systems has been of interest because of its implications on uniqueness of Gibbs measures on infinite graphs and efficient approximation of counting problems that are otherwise known to be #P hard. In the context of coloring, strong spatial mixing has been established for Kelly trees in (Ge and Stefankovic, arXiv:1102.2886v3 (2011)) when q ≥ α[superscript *] Δ + 1 where q the number of colors, Δ is the degree and α[superscript *] is the unique solution to xe[superscript -1/x] = 1. It has also been established in (Goldberg et al., SICOMP 35 (2005) 486–517) for bounded degree lattice graphs whenever q ≥ α[superscript *] Δ - β for some constant β, where Δ is the maximum vertex degree of the graph. We establish strong spatial mixing for a more general problem, namely list coloring, for arbitrary bounded degree triangle-free graphs. Our results hold for any α > α[superscript *] whenever the size of the list of each vertex v is at least αΔ(v) + β where Δ(v) is the degree of vertex v and β is a constant that only depends on α. The result is obtained by proving the decay of correlations of marginal probabilities associated with graph nodes measured using a suitably chosen error function.