Arboricity-Dependent Algorithms for Edge Coloring

Abstract: The problem of edge coloring has been extensively studied over the years. Recently, this problem has received significant attention in the dynamic setting, where we are given a dynamic graph evolving via a sequence of edge insertions and deletions and our objective is to maintain an edge coloring of the graph. Currently, it is not known whether it is possible to maintain a $(\Delta+ O(\Delta^{1 - \mu}))$-edge coloring in $\tilde{O}(1)$ update time, for any constant $\mu > 0$, where $\Delta$ is the maximum degree of the graph. In this paper, we show how to efficiently maintain a $(\Delta + O(\alpha))$-edge coloring in $\tilde O(1)$ amortized update time, where $\alpha$ is the arboricty of the graph. Thus, we answer this question in the affirmative for graphs of sufficiently small arboricity.