  Given a graph $G$ that is modified by a sequence of edge insertions and
deletions, we study the Maximum $k$-Edge Coloring problem Having access to $k$
colors, how can we color as many edges of $G$ as possible such that no two
adjacent edges share the same color? While this problem is different from
simply maintaining a $b$-matching with $b=k$, the two problems are closely
related: a maximum $k$-matching always contains a $\frac{k+1}k$-approximate
maximum $k$-edge coloring. However, maximum $b$-matching can be solved
efficiently in the static setting, whereas the Maximum $k$-Edge Coloring
problem is NP-hard and even APX-hard for $k \ge 2$.
  We present new results on both problems: For $b$-matching, we show a new
integrality gap result and for the case where $b$ is a constant, we adapt
Wajc's matching sparsification scheme~[STOC20].
  Using these as basis, we give three new algorithms for the dynamic Maximum
$k$-Edge Coloring problem: Our MatchO algorithm builds on the dynamic
$(2+\epsilon)$-approximation algorithm of Bhattacharya, Gupta, and
Mohan~[ESA17] for $b$-matching and achieves a $(2+\epsilon)\frac{k+1}
k$-approximation in $O(poly(\log n, \epsilon^{-1}))$ update time against an
oblivious adversary. Our MatchA algorithm builds on the dynamic
$8$-approximation algorithm by Bhattacharya, Henzinger, and Italiano~[SODA15]
for fractional $b$-matching and achieves a
$(8+\epsilon)\frac{3k+3}{3k-1}$-approximation in $O(poly(\log n,
\epsilon^{-1}))$ update time against an adaptive adversary. Moreover, our
reductions use the dynamic $b$-matching algorithm as a black box, so any future
improvement in the approximation ratio for dynamic $b$-matching will
automatically translate into a better approximation ratio for our algorithms.
Finally, we present a greedy algorithm that runs in $O(\Delta+k)$ update time,
while guaranteeing a $2.16$~approximation factor.
