A linear-time algorithm for $(1+\epsilon)\Delta$-edge-coloring

TL;DR

This paper introduces the first linear-time randomized algorithm for proper $(1+ ext{epsilon}) imes ext{Delta}$-edge-coloring of simple graphs, efficiently handling the entire range of maximum degrees with high probability.

Contribution

It provides a novel linear-time randomized algorithm for edge-coloring with $(1+ ext{epsilon}) imes ext{Delta}$ colors, covering all degrees $ ext{Delta} \\geq 1/ ext{epsilon}$, a previously unresolved problem.

Findings

Algorithm runs in $O(m)$ time with high probability.

First linear-time algorithm for full range of $ ext{Delta}$.

Works for edge-coloring with $2 ext{Delta}-1$ colors.

Abstract

We present a randomized algorithm that, given a constant $\epsilon > 0$, outputs a proper $(1+\epsilon)\Delta$-edge-coloring of an $m$-edge simple graph $G$ of maximum degree $\Delta \geq 1/\epsilon$ in $O(m)$ time with high probability. This is the first linear-time algorithm for this problem covering the full range of possible values of $\Delta$. Indeed, even for edge-coloring with $2\Delta - 1$ colors (i.e., meeting the "greedy" bound), no such linear-time algorithm has been previously known.