Improved bounds on the zeros of the chromatic polynomial of graphs and claw-free graphs

TL;DR

This paper improves bounds on the location of zeros of the chromatic polynomial for general, large girth, and claw-free graphs, introducing novel techniques and representations to tighten these bounds.

Contribution

It provides tighter bounds on chromatic polynomial zeros for various graph classes and introduces new methods involving partial acyclic orientations.

Findings

Zeros lie inside a disk of radius 4.25 Δ(G) for general graphs.

Radius can be reduced to 3.60 for graphs with large girth.

Radius can be reduced to 3.81 for claw-free graphs.

Abstract

We prove that for any graph $G$ the (complex) zeros of its chromatic polynomial, $\chi_G(x)$, lie inside the disk centered at $0$ of radius $4.25 \Delta(G)$, where $\Delta(G)$ denotes the maximum degree of $G$. This improves on a recent result of Jenssen, Patel and Regts, who proved a bound of $5.94\Delta(G)$. Moreover, we show that for graphs of sufficiently large girth we can replace $4.25$ by $3.60$ and for claw-free graphs we can replace $4.25$ by $3.81$. Our proofs add some substantially novel ideas to those developed by Jenssen, Patel, and Regts, while building on them. A key novel ingredient for claw-free graphs is to use a representation of the coefficients of the chromatic polynomial in terms of the number of certain partial acyclic orientations.