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, ..., 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-INJECTIVE k-EDGE-COLORING. We show that INJECTIVE 3-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. INJECTIVE 4-EDGE-COLORING remains NP-complete for cubic graphs. For any k >= 45, we show that INJECTIVE k-EDGE-COLORING remains NP-complete even for graphs of maximum degree at most 5 root 3k. In contrast with these negative results, we show that INJECTIVE k-EDGE-COLORING 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 root k/2 is injectively k-edge-colorable. (C) 2021 Elsevier B.V. All rights reserved.