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:维化的方程式表示, 每张以最大度為nG$ {Delta$}{Delta${Delta$\Delta$+1美元的颜色, 都可以使用$\Delta+1美元的颜色。 目前已知的通用图形中最快的$(Delta+1)$的彩色算法是因Sinnanon而导致的, 并按时间运行 $O(m\Sqrt{NG)$=<unk> V(G)$ 和 <unk> E(G)%美元。 使用受约束的 $\leq\Delta$, Sinnnaon的算法可以表现为$(Delta\D) 美元。 当$\Delta$(美元是恒定值) 时, 这可以改进, 因为Gabow, Nishizeki, Kariv, Lefo, 和Terada 设计的算法是用时间 $(O(Delta n) 美元) 正在运行的O=Calmax 美元 时间。</s>