The problem of edge coloring has been extensively studied over the years. The main conceptual contribution of this work is in identifying a surprisingly simple connection between the problem of $(\Delta +O(\alpha))$-edge coloring and a certain canonical graph decomposition in graphs of arboricity $\alpha$, for which efficient algorithms are known across various computational models. We first leverage such graph decompositions to provide fast $(\Delta +O(\alpha))$-edge coloring algorithms in the standard {\em static} (sequential and distributed) settings. Further, as our main technical contribution, we show how to efficiently maintain a $(\Delta +O(\alpha))$-edge coloring in the standard {\em dynamic} model. Consequently, we improve over the state-of-the-art edge coloring algorithms in these models for graphs of sufficiently small arboricity.

翻译：边染色问题多年来得到了广泛研究。本研究的主要概念贡献在于揭示了$(\Delta +O(\alpha))$边染色问题与树度数$\alpha$图中某种规范图分解之间存在令人惊讶的简单关联，且该分解在多种计算模型中已有高效算法。我们首先利用这种图分解在标准{\em静态}（顺序与分布式）场景中提供了快速$(\Delta +O(\alpha))$边染色算法。进一步地，作为主要技术贡献，我们展示了如何在标准{\em动态}模型下高效维护$(\Delta +O(\alpha))$边染色。由此，对于树度数足够小的图，我们改进了现有边染色算法在这些模型中的性能。