Arboricity-Dependent Algorithms for Edge Coloring

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.