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：摘要：本文研究有界最大度多重图 $G$ 的边着色算法，其中 $\Delta(G) = O(1)$。Shannon定理指出，任意最大度为 $\Delta$ 的多重图都能用 $\lfloor 3\Delta/2\rfloor$ 种颜色进行正常边着色。我们的主要结果包括计算此类着色的算法。我们设计了确定性随机顺序算法，其运行时间分别为 $O(n\log n)$ 和 $O(n)$。这是自Shannon原始论文中 $O(n^2)$ 算法以来的首次改进，且我们的随机算法在常数因子内达到最优。我们还开发了 $\mathsf{LOCAL}$ 计算模型中的分布式算法。具体而言，我们设计了确定性和随机化 $\mathsf{LOCAL}$ 算法，其运行时间分别为 $\tilde O(\log^5 n)$ 和 $O(\log^2 n)$。确定性顺序算法是对Gabow等人在简单图边着色中早期工作的简化扩展。其他算法采用熵压缩方法，其方式类似于作者与Bernshteyn近期关于简单图Vizing定理的工作。我们还将他们的结果推广到多重图的Vizing定理。