Bounds on the Complex Zeros of (Di)Chromatic Polynomials and Potts-Model Partition Functions

We show that there exist universal constants C(r) < ∞ such that, for all loopless graphs G of maximum degree ≤ r, the zeros (real or complex) of the chromatic polynomial P_G(q) lie in the disc |q| < C(r). Furthermore, C(r) ≤ 7.963907r. This result is a corollary of a more general result on the zeros of the Potts-model partition function Z_G(q, {v_e}) in the complex antiferromagnetic regime |1 + v_e| ≤ 1. The proof is based on a transformation of the Whitney-Tutte-Fortuin-Kasteleyn representation of Z_G(q, {v_e}) to a polymer gas, followed by verification of the Dobrushin-Kotecký-Preiss condition for nonvanishing of a polymer-model partition function. We also show that, for all loopless graphs G of second-largest degree ≤ r, the zeros of P_G(q) lie in the disc |q| < C(r) + 1. Along the way, I give a simple proof of a generalized (multivariate) Brown-Colboum conjecture on the zeros of the reliability polynomial for the special case of series-parallel graphs.