  Using the framework of advice complexity, we study the amount of knowledge
about the future that an online algorithm needs to color the edges of a graph
optimally, i.e., using as few colors as possible. For graphs of maximum degree
$\Delta$, it follows from Vizing's Theorem that $O(m\log \Delta)$ bits of
advice suffice to achieve optimality, where $m$ is the number of edges. We show
that for graphs of bounded degeneracy (a class of graphs including e.g. trees
and planar graphs), only $O(m)$ bits of advice are needed to compute an optimal
solution online, independently of how large $\Delta$ is. On the other hand, we
show that $\Omega (m)$ bits of advice are necessary just to achieve a
competitive ratio better than that of the best deterministic online algorithm
without advice. Furthermore, we consider algorithms which use a fixed number of
advice bits per edge (our algorithm for graphs of bounded degeneracy belongs to
this class of algorithms). We show that for bipartite graphs, any such
algorithm must use at least $\Omega(m\log \Delta)$ bits of advice to achieve
optimality.
