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

TL;DR

This paper establishes universal bounds on the location of zeros of chromatic and Potts-model partition functions for graphs with bounded degree, using polymer gas transformations and complex analysis techniques.

Contribution

It introduces a new bound on zeros of chromatic polynomials for graphs with maximum degree r, extending to Potts-model functions in the antiferromagnetic regime, and proves a generalized Brown-Colbourn conjecture for series-parallel graphs.

Findings

Zeros of chromatic polynomials lie in a bounded disc depending on maximum degree.

Zeros of Potts-model partition functions are similarly bounded in the complex plane.

A simple proof of a generalized Brown-Colbourn conjecture for series-parallel graphs.

Abstract

I show that there exist universal constants $C(r) < \infty$ such that, for all loopless graphs $G$ of maximum degree $\le r$, the zeros (real or complex) of the chromatic polynomial $P_G(q)$ lie in the disc $|q| < C(r)$. Furthermore, $C(r) \le 7.963906... r$. 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 $|1 + v_e| \le 1$. 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-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 $\le r$, 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-Colbourn conjecture on the zeros of the reliability polynomial for the special case of series-parallel graphs.