We consider the problem of maintaining a $(1+\epsilon)\Delta$-edge coloring in a dynamic graph $G$ with $n$ nodes and maximum degree at most $\Delta$. The state-of-the-art update time is $O_\epsilon(\text{polylog}(n))$, by Duan, He and Zhang [SODA'19] and by Christiansen [STOC'23], and more precisely $O(\log^7 n/\epsilon^2)$, where $\Delta = \Omega(\log^2 n / \epsilon^2)$. The following natural question arises: What is the best possible update time of an algorithm for this task? More specifically, \textbf{ can we bring it all the way down to some constant} (for constant $\epsilon$)? This question coincides with the \emph{static} time barrier for the problem: Even for $(2\Delta-1)$-coloring, there is only a naive $O(m \log \Delta)$-time algorithm. We answer this fundamental question in the affirmative, by presenting a dynamic $(1+\epsilon)\Delta$-edge coloring algorithm with $O(\log^4 (1/\epsilon)/\epsilon^9)$ update time, provided $\Delta = \Omega_\epsilon(\text{polylog}(n))$. As a corollary, we also get the first linear time (for constant $\epsilon$) \emph{static} algorithm for $(1+\epsilon)\Delta$-edge coloring; in particular, we achieve a running time of $O(m \log (1/\epsilon)/\epsilon^2)$. We obtain our results by carefully combining a variant of the \textsc{Nibble} algorithm from Bhattacharya, Grandoni and Wajc [SODA'21] with the subsampling technique of Kulkarni, Liu, Sah, Sawhney and Tarnawski [STOC'22].

翻译：我们考虑在动态图 $G$ 中维护 $(1+\epsilon)\Delta$-边着色问题，该图有 $n$ 个节点且最大度不超过 $\Delta$。当前最优更新时间为 $O_\epsilon(\text{polylog}(n))$，由 Duan、He 和 Zhang [SODA'19] 以及 Christiansen [STOC'23] 提出，更精确地说是 $O(\log^7 n/\epsilon^2)$，其中 $\Delta = \Omega(\log^2 n / \epsilon^2)$。由此自然产生一个问题：这类任务算法的可能最优更新时间是多少？具体而言，**我们能否将其降至某个常数**（对于恒定 $\epsilon$）？该问题与问题的*静态*时间瓶颈相吻合：即使是 $(2\Delta-1)$-着色，也仅有朴素的 $O(m \log \Delta)$ 时间算法。我们通过提出一种动态 $(1+\epsilon)\Delta$-边着色算法，以 $O(\log^4 (1/\epsilon)/\epsilon^9)$ 更新时间（当 $\Delta = \Omega_\epsilon(\text{polylog}(n))$ 时）肯定地回答了这一基本问题。作为推论，我们还首次得到（对于恒定 $\epsilon$）*静态* $(1+\epsilon)\Delta$-边着色线性时间算法；具体而言，我们实现了 $O(m \log (1/\epsilon)/\epsilon^2)$ 的运行时间。我们的结果是通过将 Bhattacharya、Grandoni 和 Wajc [SODA'21] 的 \textsc{Nibble} 算法变体与 Kulkarni、Liu、Sah、Sawhney 和 Tarnawski [STOC'22] 的子采样技术巧妙结合而获得。