Fast and Simple $(1+\epsilon)\Delta$-Edge-Coloring of Dense Graphs
<|reference_start|>Fast and Simple $(1+\epsilon)\Delta$-Edge-Coloring of Dense Graphs: Let $\epsilon \in (0, 1)$ and $n, \Delta \in \mathbb N$ be such that $\Delta = \Omega\left(\max\left\{\frac{\log n}{\epsilon},\, \left(\frac{1}{\epsilon}\log \frac{1}{\epsilon}\right)^2\right\}\right)$. Given an $n$-vertex $m$-edge simple graph $G$ of maximum degree $\Delta$, we present a randomized $O\left(m\,\log^3 \Delta\,/\,\epsilon^2\right)$-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 $\epsilon$, $n$, and $\Delta$. 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.<|reference_end|>
@article{dhawan2024fast,
    title={Fast and Simple $(1+\epsilon)\Delta$-Edge-Coloring of Dense Graphs},
    author={Abhishek Dhawan},
    journal={arXiv preprint arXiv:2408.16692},
    year={2024},
    archivePrefix={arXiv},
    eprint={2408.16692},
    primaryClass={cs.DS math.CO}
}