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 ${\chi }_G\left(x\right)$ (in $C$) lie inside the disc of radius 5.94Δ 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 ${\chi }_G\left(x\right)$ 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 $g\ge 5$ and $K_g<4$ when $g\ge 25$ 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.