Complexity and algorithms for injective edge-coloring in graphs

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