Planar graph

1-planar graph

Combinatorics

Discrete mathematics

Edge coloring

Injective function

Graph

Treewidth

Graph power

Chordal graph

Bipartite graph

Line graph

Mathematics

Pathwidth

Graph coloring

Information Processing Letters

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 53k. 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 k/2 is injectively k-edge-colorable.

Complexity and algorithms for injective edge-coloring in graphs