  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.
