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

Abstract

We show that there exist universal constants C(r) < ∞ such that, for all loopless graphs G of maximum degree [les ] r, the zeros (real or complex) of the chromatic polynomial PG(q) lie in the disc [mid ]q[mid ] < C(r). Furthermore, C(r) [les ] 7.963907r. This result is a corollary of a more general result on the zeros of the Potts-model partition function ZG(q, {ve}) in the complex antiferromagnetic regime [mid ]1 + ve[mid ] [les ] 1. The proof is based on a transformation of the Whitney–Tutte–Fortuin–Kasteleyn representation of ZG(q, {ve}) 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 [les ] r, the zeros of PG(q) lie in the disc [mid ]q[mid ] < C(r) + 1. Along the way, I give a simple proof of a generalized (multivariate) Brown–Colbourn conjecture on the zeros of the reliability polynomial for the special case of series-parallel graphs.