Near-optimal distributed edge coloring
For any fixed λ>0, colours G with (1+λ)Δ colours in time O(log n).
For any fixed positive integer s, colours G with Δ+Δ/(log Δ)s=(1+o(1))Δ colours in time O(log n+logsΔ loglog Δ).
Both results hold with probability arbitrarily close to 1 as long as Δ(G) =Ω(log1+dn), for some d > 0. The algorithm is based on the Rödl Nibble, a probabilistic strategy introduced by Vojtech Rödl. The analysis involves a certain quasi-random phenomenon involving sets at the vertices of the graph.
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