Deterministic Approximate Counting of Colorings with fewer than 2Δ Colors via Absence of Zeros

Let Δ,q≥3 be integers. We prove that there exists η≥0.002 such that if q≥(2−η)Δ, then there exists an open set U⊂C that contains the interval [0,1] such that for each w∈U and any graph G=(V,E) of maximum degree at most Δ, the partition function of the anti-ferromagnetic q-state Potts model evaluated at w does not vanish. This provides a (modest) improvement on a result of Liu, Sinclair, and Srivastava, and breaks the q=2Δ-barrier for this problem.
As a direct consequence we obtain via Barvinok's interpolation method a deterministic polynomial time algorithm to approximate the number of proper q-colorings of graphs of maximum degree at most Δ, provided q≥(2−η)Δ.