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 $O\left(m\cdot\min\left\{\Delta\cdot\log n,\sqrt{n}\right\}\right)$, and very recently to randomized $\tilde{O}\left(m\cdot n^{1/3}\right)$. A randomized $(1+\varepsilon)\Delta$-edge-coloring algorithm can be computed in $O\left(m\cdot\frac{\log^6 n}{\varepsilon^2}\right)$ time, and for large values of $\Delta$, this task requires randomized $O\left(\frac{m\cdot\log\varepsilon^{-1}}{\varepsilon^2}\right)$ time. 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 $O\left(m\cdot\frac{\log n}{\varepsilon}\right)$. A randomized variant of our algorithm has running time $O(m\cdot(\varepsilon^{-18}+\log(\varepsilon\cdot\Delta)))$. We also study edge-coloring of graphs with arboricity at most $\alpha$. A randomized computation of $(\Delta+1)$-edge-coloring requires $\tilde{O}\left(\min\{m\cdot\sqrt{n},m\cdot\Delta\}\cdot\frac{\alpha}{\Delta}\right)$ time. Deterministically, this task can be done in $O\left(m\cdot\alpha^7\cdot\log n\right)$ time. However, for large values of $\alpha$, these algorithms require super-linear time. We devise a deterministic $(\Delta+\varepsilon\alpha)$-edge-coloring algorithm with running time $O\left(\frac{m\cdot\log n}{\varepsilon^7}\right)$. A randomized version of our algorithm requires $O\left(\frac{m\cdot\log n}{\varepsilon}\right)$ expected time. Our algorithm is based on a novel two-way degree-splitting, which we devise in this paper. We believe that this technique is of independent interest.

翻译：我们研究具有最大度 $\Delta$ 的简单 $n$ 顶点 $m$ 边图的边着色问题。这是图算法中最经典和基础的问题之一。Vizing 著名定理提供了 $O(m\cdot n)$ 确定性时间内的 $(\Delta+1)$-边着色。该运行时间被改进为 $O\left(m\cdot\min\left\{\Delta\cdot\log n,\sqrt{n}\right\}\right)$，并在最近改进为随机化的 $\tilde{O}\left(m\cdot n^{1/3}\right)$。一个随机化的 $(1+\varepsilon)\Delta$-边着色算法可以在 $O\left(m\cdot\frac{\log^6 n}{\varepsilon^2}\right)$ 时间内计算得到，并且对于较大的 $\Delta$ 值，此任务需要随机化的 $O\left(\frac{m\cdot\log\varepsilon^{-1}}{\varepsilon^2}\right)$ 时间。然而，对于这一基本问题是否存在确定性的近线性时间算法是悬而未决的。我们设计了一个简单的确定性 $(1+\varepsilon)\Delta$-边着色算法，其运行时间为 $O\left(m\cdot\frac{\log n}{\varepsilon}\right)$。我们算法的随机化变体运行时间为 $O(m\cdot(\varepsilon^{-18}+\log(\varepsilon\cdot\Delta)))$。我们还研究了树密度至多为 $\alpha$ 的图的边着色。$(\Delta+1)$-边着色的随机化计算需要 $\tilde{O}\left(\min\{m\cdot\sqrt{n},m\cdot\Delta\}\cdot\frac{\alpha}{\Delta}\right)$ 时间。确定性地，此任务可以在 $O\left(m\cdot\alpha^7\cdot\log n\right)$ 时间内完成。然而，对于较大的 $\alpha$ 值，这些算法需要超线性时间。我们设计了一个确定性的 $(\Delta+\varepsilon\alpha)$-边着色算法，其运行时间为 $O\left(\frac{m\cdot\log n}{\varepsilon^7}\right)$。我们算法的随机化版本需要 $O\left(\frac{m\cdot\log n}{\varepsilon}\right)$ 期望时间。我们的算法基于一种新颖的双向度分割技术，这是我们在本文中设计的。我们相信该技术具有独立的研究价值。