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

Abstract

We prove that for any graph $G$ of maximum degree at most $Δ$, the zeros of its chromatic polynomial $χ_G(x)$ (in $\mathbb{C}$) lie inside the disc of radius $5.94 Δ$ centered at $0$. This improves on the previously best known bound of approximately $6.91Δ$. 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 $Δ$ and girth at least $g$, the zeros of its chromatic polynomial $χ_G(x)$ lie inside the disc of radius $K_g Δ$ 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.