Vizing's theorem states that any $n$-vertex $m$-edge graph of maximum degree $\Delta$ can be {\em edge colored} using at most $\Delta + 1$ different colors [Diskret.~Analiz, '64]. Vizing's original proof is algorithmic and shows that such an edge coloring can be found in $\tilde{O}(mn)$ time. This was subsequently improved to $\tilde O(m\sqrt{n})$, independently by Arjomandi [1982] and by Gabow et al.~[1985]. In this paper we present an algorithm that computes such an edge coloring in $\tilde O(mn^{1/3})$ time, giving the first polynomial improvement for this fundamental problem in over 40 years.

翻译：维津定理指出，任意具有$n$个顶点、$m$条边且最大度为$\Delta$的图，最多可使用$\Delta + 1$种不同颜色进行边着色[Diskret.~Analiz, '64]。维津的原始证明是构造性的，并表明这样的边着色可以在$\tilde{O}(mn)$时间内找到。随后，Arjomandi [1982]与Gabow等人[1985]分别独立地将时间改进至$\tilde O(m\sqrt{n})$。本文提出一种算法，可在$\tilde O(mn^{1/3})$时间内计算此类边着色，为该基础性问题提供了40多年来的首次多项式时间改进。