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 $m :=|E(G)|$. In this paper we investigate the case when $\Delta$ is constant, i.e., $\Delta = O(1)$. In this regime, the running time of Sinnamon's algorithm is $O(n^{3/2})$, which can be improved to $O(n \log n)$, as shown by Gabow, Nishizeki, Kariv, Leven, and Terada. Here we give an algorithm whose running time is only $O(n)$, which is obviously best possible. 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 $\tilde{O}(\log^5 n)$ and a randomized $\mathsf{LOCAL}$ algorithm with running time $O(\log ^2 n)$. All these results are new already for $\Delta = 4$. Although our focus is on the constant $\Delta$ regime, our results remain interesting for $\Delta$ up to $\log^{o(1)} n$. The key new ingredient in our algorithms is a novel application of the entropy compression method.

翻译：Vizing定理指出，每个最大度数为$\Delta$的图$G$都可以用$\Delta + 1$种颜色进行正常边着色。目前针对一般图最快的已知$(\Delta+1)$-边着色算法由Sinnamon提出，运行时间为$O(m\sqrt{n})$，其中$n :=|V(G)|$，$m :=|E(G)|$。本文研究$\Delta$为常数的情况，即$\Delta = O(1)$。在此范围内，Sinnamon算法的运行时间为$O(n^{3/2})$，而Gabow、Nishizeki、Kariv、Leven和Terada已将其改进至$O(n \log n)$。我们在此给出一个运行时间仅为$O(n)$的算法，这显然是最优的。此外，我们还针对分布式计算的$\mathsf{LOCAL}$模型开发了新的$(\Delta+1)$-边着色算法：设计了一个运行时间为$\tilde{O}(\log^5 n)$的确定性$\mathsf{LOCAL}$算法，以及一个运行时间为$O(\log^2 n)$的随机化$\mathsf{LOCAL}$算法。这些结果即使对于$\Delta = 4$的情况也是全新的。尽管我们的研究重点在于常数$\Delta$的范围，但结果对于$\Delta$高达$\log^{o(1)} n$的情况仍具意义。我们算法中的关键新要素是熵压缩方法的新颖应用。