We study the edge-coloring problem in simple $n$-vertex $m$-edge graphs with maximum degree $\Delta$. This is one of the most classical and fundamental graph-algorithmic problems. Vizing's celebrated theorem provides $(\Delta+1)$-edge-coloring in $O(m\cdot n)$ deterministic time. This running time was improved to $O\left(m\cdot\min\left\{\Delta\cdot\log n, \sqrt{n}\right\}\right)$. It is also well-known that $3\left\lceil\frac{\Delta}{2}\right\rceil$-edge-coloring can be computed in $O(m\cdot\log\Delta)$ time deterministically. Duan et al. devised a randomized $(1+\varepsilon)\Delta$-edge-coloring algorithm with running time $O\left(m\cdot\frac{\log^6 n}{\varepsilon^2}\right)$. It was however open if there exists a deterministic near-linear time algorithm for this basic problem. We devise a simple deterministic $(1+\varepsilon)\Delta$-edge-coloring algorithm with running time $O\left(m\cdot\frac{\log n}{\varepsilon}\right)$. We also devise a randomized $(1+\varepsilon)\Delta$-edge-coloring algorithm with running time $O(m\cdot(\varepsilon^{-18}+\log(\varepsilon\cdot\Delta)))$. For $\varepsilon\geq\frac{1}{\log^{1/18}\Delta}$, this running time is $O(m\cdot\log\Delta)$.

翻译：我们研究最大度为$\Delta$的简单$n$顶点$m$边图中的边染色问题。这是最经典且基础的图算法问题之一。维津定理提供了$O(m\cdot n)$确定性时间内的$(\Delta+1)$-边染色算法。该运行时间已被改进至$O\left(m\cdot\min\left\{\Delta\cdot\log n, \sqrt{n}\right\}\right)$。此外，众所周知，$3\left\lceil\frac{\Delta}{2}\right\rceil$-边染色可在$O(m\cdot\log\Delta)$时间内确定性计算。段等人设计了一种随机化$(1+\varepsilon)\Delta$-边染色算法，运行时间为$O\left(m\cdot\frac{\log^6 n}{\varepsilon^2}\right)$。然而，是否存在针对该基本问题的确定性近线性时间算法此前仍是未解难题。我们设计了一种简单的确定性$(1+\varepsilon)\Delta$-边染色算法，运行时间为$O\left(m\cdot\frac{\log n}{\varepsilon}\right)$。我们还设计了一种随机化$(1+\varepsilon)\Delta$-边染色算法，运行时间为$O(m\cdot(\varepsilon^{-18}+\log(\varepsilon\cdot\Delta)))$。当$\varepsilon\geq\frac{1}{\log^{1/18}\Delta}$时，该运行时间为$O(m\cdot\log\Delta)$。