  In a proper edge-coloring the edges of every color form a matching. A
matching is induced if the end-vertices of its edges induce a matching. A
strong edge-coloring is an edge-coloring in which the edges of every color form
an induced matching. We consider intermediate types of edge-colorings, where
edges of some colors are allowed to form matchings, and the remaining form
induced matchings. Our research is motivated by the conjecture proposed in a
recent paper of Gastineau and Togni on S-packing edge-colorings (On S-packing
edge-colorings of cubic graphs, Discrete Appl. Math. 259 (2019), 63-75)
asserting that by allowing three additional induced matchings, one is able to
save one matching color. We prove that every graph with maximum degree 3 can be
decomposed into one matching and at most 8 induced matchings, and two matchings
and at most 5 induced matchings. We also show that if a graph is in class I,
the number of induced matchings can be decreased by one, hence confirming the
above-mentioned conjecture for class I graphs.
