Chromatic Polynomials, Potts Models and All That

TL;DR

This paper explores the connections between the Potts model, chromatic polynomials, and graph properties, presenting new results on the distribution of chromatic zeros and bounds related to graph degree.

Contribution

It introduces two recent results: a family of planar graphs with dense chromatic zeros and a universal upper bound on zeros based on maximum degree.

Findings

Chromatic zeros are dense in the complex plane for certain planar graphs.

A universal upper bound on chromatic polynomial zeros related to maximum degree.

Pedagogical overview of Potts models and graph polynomials.

Abstract

The q-state Potts model can be defined on an arbitrary finite graph, and its partition function encodes much important information about that graph, including its chromatic polynomial, flow polynomial and reliability polynomial. The complex zeros of the Potts partition function are of interest both to statistical mechanicians and to combinatorists. I give a pedagogical introduction to all these problems, and then sketch two recent results: (a) Construction of a countable family of planar graphs whose chromatic zeros are dense in the whole complex q-plane except possibly for the disc |q-1| < 1. (b) Proof of a universal upper bound on the q-plane zeros of the chromatic polynomial (or antiferromagnetic Potts-model partition function) in terms of the graph's maximum degree.