  Let $G=(V,E)$ be a graph. A (proper) $k$-edge-coloring is a coloring of the
edges of $G$ such that any pair of edges sharing an endpoint receive distinct
colors. A classical result of Vizing ensures that any simple graph $G$ admits a
$(\Delta(G)+1)$-edge coloring where $\Delta(G)$ denotes the maximum degreee of
$G$. Recently, Cabello raised the following question: given two graphs
$G_1,G_2$ of maximum degree $\Delta$ on the same set of vertices $V$, is it
possible to edge-color their (edge) union with $\Delta+2$ colors in such a way
the restriction of $G$ to respectively the edges of $G_1$ and the edges of
$G_2$ are edge-colorings? More generally, given $\ell$ graphs, how many colors
do we need to color their union in such a way the restriction of the coloring
to each graph is proper?
  In this short note, we prove that we can always color the union of the graphs
$G_1,\ldots,G_\ell$ of maximum degree $\Delta$ with $\Omega(\sqrt{\ell} \cdot
\Delta)$ colors and that there exist graphs for which this bound is tight up to
a constant multiplicative factor. Moreover, for two graphs, we prove that at
most $\frac 32 \Delta +4$ colors are enough which is, as far as we know, the
best known upper bound.
