Let $\Delta,q\geq 3$ be integers. We prove that there exists $\eta\geq 0.002$ such that if $q\geq (2-\eta)\Delta$, then there exists an open set $\mathcal{U}\subset \mathbb{C}$ that contains the interval $[0,1]$ such that for each $w\in \mathcal{U}$ and any graph $G=(V,E)$ of maximum degree at most $\Delta$, the partition function of the anti-ferromagnetic $q$-state Potts model evaluated at $w$ does not vanish. This provides a (modest) improvement on a result of Liu, Sinclair, and Srivastava, and breaks the $q=2\Delta$-barrier for this problem. As a direct consequence we obtain via Barvinok's interpolation method a deterministic polynomial time algorithm to approximate the number of proper $q$-colorings of graphs of maximum degree at most $\Delta$, provided $q\geq (2-\eta)\Delta$.

翻译：令$\Delta,q\geq 3$为整数。我们证明存在$\eta\geq 0.002$，使得当$q\geq (2-\eta)\Delta$时，存在包含区间$[0,1]$的开集$\mathcal{U}\subset \mathbb{C}$，满足对任意$w\in \mathcal{U}$及任意最大度不超过$\Delta$的图$G=(V,E)$，反铁磁$q$-态Potts模型在$w$处的配分函数均非零。该结果对Liu、Sinclair与Srivastava的研究实现了（适度）改进，并突破了该问题中$q=2\Delta$的理论界限。通过Barvinok插值法，我们直接推导出一个确定性多项式时间算法，可在$q\geq (2-\eta)\Delta$条件下近似计算最大度不超过$\Delta$的图的真$q$-着色方案数目。