Vizing's theorem states that every graph $G$ of maximum degree $\Delta$ can be properly edge-colored using $\Delta + 1$ colors. The fastest currently known $(\Delta+1)$-edge-coloring algorithm for general graphs is due to Sinnamon and runs in time $O(m\sqrt{n})$, where $n = |V(G)|$ and $m =|E(G)|$. Using the bound $m \leq \Delta n/2$, the running time of Sinnamon's algorithm can be expressed as $O(\Delta n^{3/2})$. In the regime when $\Delta$ is considerably smaller than $n$ (for instance, when $\Delta$ is a constant), this can be improved, as Gabow, Nishizeki, Kariv, Leven, and Terada designed an algorithm with running time $O(\Delta m \log n) = O(\Delta^2 n \log n)$. Here we give an algorithm whose running time is only linear in $n$ (which is obviously best possible) and polynomial in $\Delta$. We also develop new algorithms for $(\Delta+1)$-edge-coloring in the $\mathsf{LOCAL}$ model of distributed computation. Namely, we design a deterministic $\mathsf{LOCAL}$ algorithm with running time $\mathsf{poly}(\Delta, \log\log n) \log^5 n$ and a randomized $\mathsf{LOCAL}$ algorithm with running time $\mathsf{poly}(\Delta) \log^2 n$. The key new ingredient in our algorithms is a novel application of the entropy compression method.

翻译：Vizing定理指出，任意最大度为$\Delta$的图$G$都可以用$\Delta+1$种颜色进行正常边着色。目前已知最快的通用图$(\Delta+1)$-边着色算法由Sinnamon提出，其运行时间为$O(m\sqrt{n})$，其中$n = |V(G)|$，$m = |E(G)|$。利用$m \leq \Delta n/2$这一界限，Sinnamon算法的运行时间可表示为$O(\Delta n^{3/2})$。当$\Delta$远小于$n$时（例如$\Delta$为常数），该结果可进一步改进：Gabow、Nishizeki、Kariv、Leven和Terada设计的算法运行时间为$O(\Delta m \log n) = O(\Delta^2 n \log n)$。本文提出的算法运行时间在$n$上仅为线性（这显然是最优的），在$\Delta$上为多项式级。我们还针对分布式计算$\mathsf{LOCAL}$模型开发了新的$(\Delta+1)$-边着色算法：具体地，我们设计了一个运行时间为$\mathsf{poly}(\Delta, \log\log n) \log^5 n$的确定性$\mathsf{LOCAL}$算法，以及一个运行时间为$\mathsf{poly}(\Delta) \log^2 n$的随机化$\mathsf{LOCAL}$算法。我们算法中的关键创新在于对熵压缩方法的新颖应用。