Regions without complex zeros for chromatic polynomials on graphs with bounded degree

TL;DR

This paper establishes bounds on the complex zeros of chromatic polynomials for graphs with bounded degree, improving previous results and providing stronger conditions for certain graph classes.

Contribution

It proves zero-free regions for chromatic polynomials of graphs with bounded degree, refining earlier bounds and extending results to graphs without triangle-free vertices.

Findings

Chromatic polynomial zeros are absent for large enough complex values depending on maximum degree.

Improved bounds on zero-free regions compared to prior work by Sokal and Borgs.

Stronger conditions for graphs with no triangle-free vertices.

Abstract

We prove that the chromatic polynomial $P_\mathbb{G}(q)$ of a finite graph $\mathbb{G}$ of maximal degree $\D$ is free of zeros for $\card q\ge C^*(\D)$ with $$ C^*(\D) = \min_{0<x<2^{1\over \D}-1} {(1+x)^{\D-1}\over x [2-(1+x)^\D]} $$ This improves results by Sokal (2001) and Borgs (2005). Furthermore, we present a strengthening of this condition for graphs with no triangle-free vertices.