Fast and Simple $(1+ε)Δ$-Edge-Coloring of Dense Graphs
Abhishek Dhawan
TL;DR
We address fast, high-probability $(1+\varepsilon)\Delta$-edge-coloring for dense graphs by combining a flagging strategy with a shifting variant of Vizing chains. The two-stage approach colors most edges in Stage 1 using disjoint palettes and short augmenting structures, then completes the coloring in Stage 2 with a folklore $(2+\varepsilon)\Delta$-edge-coloring routine. The main technical contribution is showing that the set of flagged edges forms a subgraph with maximum degree $O(\varepsilon\Delta)$ w.h.p., enabling a final colouring in near-linear time: $O\left(m\log^3\Delta/\varepsilon^2\right)$. The method achieves high-probability correctness under $\Delta = \Omega\left(\max\left\{\frac{\log n}{\varepsilon},\left(\frac{1}{\varepsilon}\log\frac{1}{\varepsilon}\right)^2\right\}\right)$ and improves upon prior results for a broad range of parameters, while remaining simple to implement. This advances practical edge-coloring for dense graphs and complements existing dynamic and two-stage approaches.
Abstract
Let $ε\in (0, 1)$ and $n, Δ\in \mathbb N$ be such that $Δ= Ω\left(\max\left\{\frac{\log n}ε,\, \left(\frac{1}ε\log \frac{1}ε\right)^2\right\}\right)$. Given an $n$-vertex $m$-edge simple graph $G$ of maximum degree $Δ$, we present a randomized $O\left(m\,\log^3 Δ\,/\,ε^2\right)$-time algorithm that computes a proper $(1+ε)Δ$-edge-coloring of $G$ with high probability. This improves upon the best known results for a wide range of the parameters $ε$, $n$, and $Δ$. Our approach combines a flagging strategy from earlier work of the author with a shifting procedure employed by Duan, He, and Zhang for dynamic edge-coloring. The resulting algorithm is simple to implement and may be of practical interest.