In this paper, we consider algorithms for edge-coloring multigraphs $G$ of bounded maximum degree, i.e., $\Delta(G) = O(1)$. Shannon's theorem states that any multigraph of maximum degree $\Delta$ can be properly edge-colored with $\lfloor 3\Delta/2\rfloor$ colors. Our main results include algorithms for computing such colorings. We design deterministic and randomized sequential algorithms with running time $O(n\log n)$ and $O(n)$, respectively. This is the first improvement since the $O(n^2)$ algorithm in Shannon's original paper, and our randomized algorithm is optimal up to constant factors. We also develop distributed algorithms in the $\mathsf{LOCAL}$ model of computation. Namely, we design deterministic and randomized $\mathsf{LOCAL}$ algorithms with running time $\tilde O(\log^5 n)$ and $O(\log^2n)$, respectively. The deterministic sequential algorithm is a simplified extension of earlier work of Gabow et al. in edge-coloring simple graphs. The other algorithms apply the entropy compression method in a similar way to recent work by the author and Bernshteyn, where the authors design algorithms for Vizing's theorem for simple graphs. We also extend their results to Vizing's theorem for multigraphs.

翻译：本文研究最大度有界（即 $\Delta(G) = O(1)$）的多重图 $G$ 的边着色算法。Shannon 定理指出，任何最大度为 $\Delta$ 的多重图均可用 $\lfloor 3\Delta/2\rfloor$ 种颜色进行正常边着色。我们的主要成果包括计算此类着色的算法：分别设计了运行时间为 $O(n\log n)$ 和 $O(n)$ 的确定性与随机化顺序算法。这是自 Shannon 原始论文中 $O(n^2)$ 算法以来的首次改进，且我们的随机化算法在常数因子意义下达到最优。此外，我们在 $\mathsf{LOCAL}$ 计算模型下开发了分布式算法，具体设计了运行时间为 $\tilde O(\log^5 n)$ 和 $O(\log^2 n)$ 的确定性与随机化 $\mathsf{LOCAL}$ 算法。确定性顺序算法是 Gabow 等人关于简单图边着色早期工作的简化扩展。其他算法采用熵压缩方法，其应用方式与作者及 Bernshteyn 近期关于简单图 Vizing 定理的工作类似。我们将其实结果进一步推广至多重图的 Vizing 定理。