  An injective $k$-edge-coloring of a graph $G$ is an assignment of colors,
i.e. integers in $\{1, \ldots , k\}$, to the edges of $G$ such that any two
edges each incident with one distinct endpoint of a third edge, receive
distinct colors. The problem of determining whether such a $k$-coloring exists
is called k-INJECTIVE EDGE-COLORING. We show that 3-INJECTIVE EDGE-COLORING is
NP-complete, even for triangle-free cubic graphs, planar subcubic graphs of
arbitrarily large girth, and planar bipartite subcubic graphs of girth~6.
4-INJECTIVE EDGE-COLORING remains NP-complete for cubic graphs. For any $k\geq
45$, we show that k-INJECTIVE EDGE-COLORING remains NP-complete even for graphs
of maximum degree at most $5\sqrt{3k}$. In contrast with these negative
results, we show that \InjPbName{k} is linear-time solvable on graphs of
bounded treewidth. Moreover, we show that all planar bipartite subcubic graphs
of girth at least~16 are injectively $3$-edge-colorable. In addition, any graph
of maximum degree at most $\sqrt{k/2}$ is injectively $k$-edge-colorable.
