Online Edge Coloring is (Nearly) as Easy as Offline
CoRR(2024)
摘要
The classic theorem of Vizing (Diskret. Analiz.'64) 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 (IPL'92) 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., 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).
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