A Fast Distributed Algorithm for $(\Delta + 1)$-Edge-Coloring

TL;DR

This paper introduces a deterministic distributed algorithm that efficiently finds a proper $(+1)$-edge-coloring of graphs in the LOCAL model, matching Vizing's bound with improved round complexity.

Contribution

It presents the first nontrivial distributed algorithm for $(+1)$-edge-coloring using only +1 colors, inspired by recent theoretical advances.

Findings

Algorithm runs in ^{poly}(, \,log n) rounds

Achieves proper edge-coloring with +1 colors in distributed setting

First to match Vizing's bound in a distributed algorithm

Abstract

We present a deterministic distributed algorithm in the LOCAL model that finds a proper $(\Delta + 1)$-edge-coloring of an $n$-vertex graph of maximum degree $\Delta$ in $\mathrm{poly}(\Delta, \log n)$ rounds. This is the first nontrivial distributed edge-coloring algorithm that uses only $\Delta+1$ colors (matching the bound given by Vizing's theorem). Our approach is inspired by the recent proof of the measurable version of Vizing's theorem due to Greb\'ik and Pikhurko.