Bounds On The Complex Zeros Of (Di)Chromatic Polynomials And Potts-Model Partition Functions

I 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 PG(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 ZG(q, {ve}) in the complex antiferromagnetic regime |1 + ve | ≤ 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–Koteck´y–Preiss condition for nonvanishing of a polymer-model partition function. I also show that, for all loopless graphs G of second-largest degree ≤ r, the zeros of PG(q) lie in the disc |q | < 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. KEY WORDS: Graph, maximum degree, second-largest degree, chromatic polynomial, dichromatic polynomial, Whitney rank function, Tutte polynomial, reliability