  Given an undirected graph $G$ and integers $c$ and $k$, the Maximum
Edge-Colorable Subgraph problem asks whether we can delete at most $k$ edges in
$G$ to obtain a graph that has a proper edge coloring with at most $c$ colors.
We show that Maximum Edge-Colorable Subgraph admits, for every fixed $c$, a
linear-size problem kernel when parameterized by the edge deletion distance of
$G$ to a graph with maximum degree $c-1$. This parameterization measures the
distance to instances that, due to Vizing's famous theorem, are trivial
yes-instances. For $c\le 4$, we also provide a linear-size kernel for the same
parameterization for Multi Strong Triadic Closure, a related edge coloring
problem with applications in social network analysis. We provide further
results for Maximum Edge-Colorable Subgraph parameterized by the vertex
deletion distance to graphs where every component has order at most $c$ and for
the list-colored versions of both problems.
