Towards the locality of Vizing's theorem
Abstract: Vizing showed that it suffices to color the edges of a simple graph using Δ + 1 colors, where Δ is the maximum degree of the graph. However, up to this date, no efficient distributed edge-coloring algorithm is known for obtaining such coloring, even for constant degree graphs. The current algorithms that get closest to this number of colors are the randomized (Δ + Θ(√Δ))-edge-coloring algorithm that runs in (n) rounds by Chang et al. [SODA 2018] and the deterministic (Δ + (n))-edge-coloring algorithm that runs in (Δ, logn) rounds by Ghaffari et al. [STOC 2018].We present two distributed edge-coloring algorithms that run in (Δ,logn) rounds. The first algorithm, with randomization, uses only Δ+2 colors. The second algorithm is a deterministic algorithm that uses Δ+ O(logn/ loglogn) colors. Our approach is to reduce the distributed edge-coloring problem into an online and restricted version of balls-into-bins problem. If ℓ is the maximum load of the bins, our algorithm uses Δ + 2ℓ − 1 colors. We show how to achieve ℓ = 1 with randomization and ℓ = O(logn / loglogn) without randomization.
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