Online Edge Coloring is (Nearly) as Easy as Offline

TL;DR

This paper proves that online edge coloring can be achieved with nearly as few colors as offline methods in the most general adversarial setting, resolving a long-standing conjecture.

Contribution

It establishes the first affirmative solution to the longstanding conjecture for online edge coloring in adversarial settings and extends to list and local edge coloring.

Findings

Online edge coloring with nearly Δ colors in adversarial settings

Resolution of the longstanding conjecture by Bar-Noy et al.

Extension to list and local edge coloring scenarios

Abstract

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).