Online Edge Coloring Is (Nearly) as Easy as Offline

.The classic theorem of Vizing [Diskret. Analiz., 3 (1964), pp. 25–30] asserts that any graph of maximum degree \(Δ\) can be edge colored (offline) using no more than \(Δ+1\) colors (with \(Δ\) being a trivial lower bound). In the online setting, Bar-Noy, Motwani, and Naor [Inform. Process. Lett., 44 (1992), pp. 251–253] conjectured that a \((1+o(1))Δ\)-edge-coloring can be computed online in \(n\)-vertex graphs of maximum degree \(Δ =ω (log n)\). 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., Aggarwal et al. [Proceedings of FOCS, 2003, pp. 502–512], Bahmani, Mehta, and Motwani (SODA'10), Cohen, Peng, and Wajc [Proceedings of FOCS, 2019, pp. 1–25], Bhattacharya, Grandoni, and Wajc [Proceedings of SODA, 2021, pp. 2830–2842], Kulkarni et al. [Proceedings of STOC, 2022, pp. 2958–2977]). 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. Combin. Theory Ser. A, 73 (1996), pp. 1–59] and of the recent "local" edge coloring result of Christiansen [Proceedings of STOC, 2023, pp. 1013–1026].Keywordsedge coloringmartingalesonline algorithmsrandomized agorithmsMSC codes05C1505C8560G4268W2768W20