Fast and simple (1 + ε)Δ-edge-coloring of dense graphs

Abstract

Let $\epsilon \in (0,1)$ and $n,\Delta \in N$ be such that $\Delta =\Omega (\max \{\frac{\log n}{\epsilon },{(\frac{1}{\epsilon }\log \frac{1}{\epsilon })}^2\})$. Given an n-vertex m-edge simple graph G of maximum degree Δ, we present a randomized $O(m{\log }^3\Delta /{\epsilon }^2)$-time algorithm that computes a proper $(1+\epsilon )\Delta$-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.