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.

翻译：我们证明，对于任意最大度不超过$\Delta$的图$G$，其色多项式$\chi_G(x)$（在$\mathbb{C}$中）的零点均位于以$0$为中心、半径为$5.94 \Delta$的圆盘内。这一结果改进了先前已知的最佳界（约为$6.91\Delta$）。对于高围长的图，我们也得到了改进的界。我们证明，对于每个$g$，存在常数$K_g$，使得对于任意最大度不超过$\Delta$且围长至少为$g$的图$G$，其色多项式$\chi_G(x)$的零点均位于以$0$为中心、半径为$K_g \Delta$的圆盘内，其中$K_g$是某个优化问题的解。特别地，当$g \geq 5$时$K_g < 5$，当$g \geq 25$时$K_g < 4$，且当$g \to \infty$时$K_g$趋近于约$3.86$。证明的关键在于Whitney的经典定理，该定理使我们能够将图$G$的色多项式与$G$中所谓无断路森林的生成函数联系起来。我们还为$G$中所有森林的生成函数（即树状气体的配分函数）建立了一个无零点圆盘，这一结果可能具有独立的研究意义。