Vizing's theorem asserts the existence of a {$(\Delta+1)$-edge coloring} for any graph $G$, where $\Delta = \Delta(G)$ denotes the maximum degree of $G$. Several polynomial time $(\Delta+1)$-edge coloring algorithms are known, and the state-of-the-art running time (up to polylogarithmic factors) is $\tilde{O}(\min\{m \cdot \sqrt{n}, m \cdot \Delta\})$, by Gabow et al.\ from 1985, where $n$ and $m$ denote the number of vertices and edges in the graph, respectively. (The $\tilde{O}$ notation suppresses polylogarithmic factors.) Recently, Sinnamon shaved off a polylogarithmic factor from the time bound of Gabow et al. The {arboricity} $\alpha = \alpha(G)$ of a graph $G$ is the minimum number of edge-disjoint forests into which its edge set can be partitioned, and it is a measure of the graph's ``uniform density''. While $\alpha \le \Delta$ in any graph, many natural and real-world graphs exhibit a significant separation between $\alpha$ and $\Delta$. In this work we design a $(\Delta+1)$-edge coloring algorithm with a running time of $\tilde{O}(\min\{m \cdot \sqrt{n}, m \cdot \Delta\})\cdot \frac{\alpha}{\Delta}$, thus improving the longstanding time barrier by a factor of $\frac{\alpha}{\Delta}$. In particular, we achieve a near-linear runtime for bounded arboricity graphs (i.e., $\alpha = \tilde{O}(1)$) as well as when $\alpha = \tilde{O}(\frac{\Delta}{\sqrt{n}})$. Our algorithm builds on Sinnamon's algorithm, and can be viewed as a density-sensitive refinement of it.

翻译：Vizing定理断言任何图$G$均存在{$(\Delta+1)$-边染色}，其中$\Delta = \Delta(G)$表示$G$的最大度数。目前已知多种多项式时间的$(\Delta+1)$-边染色算法，其最优运行时间（忽略多对数因子）为$\tilde{O}(\min\{m \cdot \sqrt{n}, m \cdot \Delta\})$（其中$n$和$m$分别表示图的顶点数和边数），该结果由Gabow等人于1985年提出（符号$\tilde{O}$表示忽略多对数因子）。近期Sinnamon将Gabow等人时间界限中的多对数因子进一步压缩。图$G$的树状度$\alpha = \alpha(G)$定义为将边集划分为边不交森林的最小数目，是衡量图“均匀密度”的指标。虽然任意图中均有$\alpha \le \Delta$，但许多自然图与真实世界图在$\alpha$与$\Delta$之间存在显著分离。本文设计了一种运行时间为$\tilde{O}(\min\{m \cdot \sqrt{n}, m \cdot \Delta\})\cdot \frac{\alpha}{\Delta}$的$(\Delta+1)$-边染色算法，从而将长期存在的运行时间界限改进了$\frac{\alpha}{\Delta}$因子。特别地，对于有界树状度图（即$\alpha = \tilde{O}(1)$）以及$\alpha = \tilde{O}(\frac{\Delta}{\sqrt{n}})$的情形，我们实现了近线性运行时间。本算法基于Sinnamon的工作，可视为对其算法的密度敏感改进。