Improved bounds for the zeros of the chromatic polynomial via Whitney's Broken Circuit Theorem

TL;DR

This paper improves bounds on the location of zeros of the chromatic polynomial for graphs with bounded degree and girth, using Whitney's Broken Circuit Theorem and analyzing related generating functions.

Contribution

It provides new, tighter bounds on chromatic polynomial zeros for graphs with maximum degree and girth constraints, extending previous results.

Findings

Zeros lie inside a disc of radius 5.94Δ for graphs with maximum degree Δ.

Improved bounds for graphs with high girth, with K_g < 5 for g ≥ 5.

K_g approaches approximately 3.86 as girth g tends to infinity.

Abstract

We prove that for any graph $G$ of maximum degree at most $\Delta$, the zeros of its chromatic polynomial $\chi_G(x)$ (in $\mathbb{C}$) lie inside the disc of radius $5.94 \Delta$ centered at $0$. This improves on the previously best known bound of approximately $6.91\Delta$. We also obtain improved bounds for graphs of high girth. We prove that for every $g$ there is a constant $K_g$ such that for any graph $G$ of maximum degree at most $\Delta$ and girth at least $g$, the zeros of its chromatic polynomial $\chi_G(x)$ lie inside the disc of radius $K_g \Delta$ centered at $0$, where $K_g$ is the solution to a certain optimization problem. In particular, $K_g < 5$ when $g \geq 5$ and $K_g < 4$ when $g \geq 25$ and $K_g$ tends to approximately $3.86$ as $g \to \infty$. Key to the proof is a classical theorem of Whitney which allows us to relate the chromatic polynomial of a graph $G$ to the generating function of so-called broken-circuit-free forests in $G$. We also establish a zero-free disc for the generating function of all forests in $G$ (aka the partition function of the arboreal gas) which may be of independent interest.