Improved bounds for zeros of the chromatic polynomial on bounded degree graphs

TL;DR

This paper establishes new bounds on the location of zeros of chromatic polynomials for graphs with bounded degree, improving previous bounds and considering high girth graphs, with implications for the Ising model.

Contribution

It provides improved bounds for zeros of chromatic polynomials in bounded degree graphs, especially for high girth graphs, advancing understanding of graph coloring properties.

Findings

Zeros lie outside disk of radius 5.02Δ, improving previous 6.91Δ bound.

For high girth graphs, zeros lie outside disk with radius approaching 3.72Δ.

Enhanced bounds on Fisher zeros of the Ising model partition function.

Abstract

We prove that for any graph $G$ of maximum degree at most $\Delta$, the zeros of its chromatic polynomial $\chi_G(z)$ (in $\mathbb{C}$) lie outside the disk of radius $5.02 \Delta$ centered at $0$. This improves on the previously best known bound of approximately $6.91\Delta$. In the case of graphs of high girth we can improve this. 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 $\chi_G(z)$ lie outside the disk of radius $K_g \Delta$ centered at $0$ where $K_g \to 1 + e \approx 3.72$ as $g \to \infty$. Finally, we give improved bounds on the Fisher zeros of the partition function of the Ising model.