On the zero-free region for the chromatic polynomial of graphs with maximum degree $\Delta$ and girth $g$

TL;DR

This paper establishes improved bounds for the zero-free region of the chromatic polynomial of graphs with given maximum degree and girth, enhancing previous results especially for small degrees.

Contribution

It provides new bounds for the zero-free region of chromatic polynomials for graphs with specified maximum degree and girth, improving upon prior bounds for small degrees.

Findings

Enhanced zero-free bounds compared to previous work

Bounds coincide with existing results as degree tends to infinity

Improved estimates for graphs with small maximum degree

Abstract

The purpose of the present paper is to provide, for all pairs of integers $(\Delta,g)$ with $\D\ge 3$ and $g\ge 3$, a positive number $C(\Delta, g)$ such that chromatic polynomial $P_G(q)$ of a graph $G$ with maximum degree $\Delta$ and finite girth $g$ is free of zero if $|q|\ge C(\Delta, g)$. Our bounds enlarge the zero-free region in the complex plane of $P_G(q)$ in comparison to previous bounds. In particular, for small values of $\D$ our estimates yield a sensible improvement on the bounds recently obtained by Jenssen, Patel and Regts in \cite{JPR}, while they coincide with those of \cite{JPR} when $\Delta\to \infty$.