Complexity and algorithms for injective edge-coloring in graphs
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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 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≥ 45, we show that k-INJECTIVE EDGE-COLORING remains NP-complete even for graphs of maximum degree at most 5√(3k). In contrast with these negative results, we show that 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 √(k/2) is injectively k-edge-colorable.
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