We prove that for any graph $G$ the (complex) zeros of its chromatic polynomial, $χ_G(x)$, lie inside the disk centered at $0$ of radius $4.25 Δ(G)$, where $Δ(G)$ denotes the maximum degree of $G$. This improves on a recent result of Jenssen, Patel and Regts, who proved a bound of $5.94Δ(G)$. Moreover, we show that for graphs of sufficiently large girth we can replace $4.25$ by $3.60$ and for claw-free graphs we can replace $4.25$ by $3.81$. Our proofs add some substantially novel ideas to those developed by Jenssen, Patel, and Regts, while building on them. A key novel ingredient for claw-free graphs is to use a representation of the coefficients of the chromatic polynomial in terms of the number of certain partial acyclic orientations.

翻译：我们证明，对于任意图$G$，其色多项式$χ_G(x)$的（复）零点位于以$0$为中心、半径为$4.25 Δ(G)$的圆盘内，其中$Δ(G)$表示$G$的最大度。这一结果改进了Jenssen、Patel和Regts最近证明的$5.94Δ(G)$界。此外，我们证明对于充分大围长的图，可将常数$4.25$改进为$3.60$；对于爪形自由图，可将常数$4.25$改进为$3.81$。我们的证明在Jenssen、Patel和Regts所发展的方法基础上，引入了若干本质新颖的思想。针对爪形自由图证明的关键创新要素，在于利用色多项式系数与特定部分无环定向数目间的表示关系。