The classic theorem of Vizing (Diskret. Analiz.'64) asserts that any graph of maximum degree $\Delta$ can be edge colored (offline) using no more than $\Delta+1$ colors (with $\Delta$ being a trivial lower bound). In the online setting, Bar-Noy, Motwani and Naor (IPL'92) conjectured that a $(1+o(1))\Delta$-edge-coloring can be computed online in $n$-vertex graphs of maximum degree $\Delta=\omega(\log n)$. Numerous algorithms made progress on this question, using a higher number of colors or assuming restricted arrival models, such as random-order edge arrivals or vertex arrivals (e.g., AGKM FOCS'03, BMM SODA'10, CPW FOCS'19, BGW SODA'21, KLSST STOC'22). In this work, we resolve this longstanding conjecture in the affirmative in the most general setting of adversarial edge arrivals. We further generalize this result to obtain online counterparts of the list edge coloring result of Kahn (J. Comb. Theory. A'96) and of the recent "local" edge coloring result of Christiansen (STOC'23).

翻译：Vizing（Diskret. Analiz.'64）的经典定理断言，任何最大度为 $\Delta$ 的图都可以（离线）使用至多 $\Delta+1$ 种颜色进行边着色（其中 $\Delta$ 是平凡下界）。在线场景下，Bar-Noy、Motwani 和 Naor（IPL'92）猜想：在最大度 $\Delta=\omega(\log n)$ 的 $n$ 顶点图中，可以在线计算一个 $(1+o(1))\Delta$ 边着色。众多算法在该问题上取得了进展，但通常使用更多颜色或假设受限的到达模型，例如随机顺序边到达或顶点到达（例如，AGKM FOCS'03、BMM SODA'10、CPW FOCS'19、BGW SODA'21、KLSST STOC'22）。在本工作中，我们在对抗性边到达的最一般设定下，肯定地解决了这一长期存在的猜想。我们进一步推广该结果，获得了 Kahn（J. Comb. Theory. A'96）的列表边着色定理以及 Christiansen（STOC'23）最近的“局部”边着色定理的在线版本。