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

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.