Complexity and algorithms for injective edge-coloring in graphs

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.