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Complexity and algorithms for injective edge-coloring in graphs

Abstract : 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 √ 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 k/2 is injectively k-edge-colorable.
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https://hal.archives-ouvertes.fr/hal-03201544
Contributor : Florent Foucaud Connect in order to contact the contributor
Submitted on : Monday, April 19, 2021 - 8:40:09 AM
Last modification on : Tuesday, January 4, 2022 - 6:17:10 AM
Long-term archiving on: : Tuesday, July 20, 2021 - 6:15:09 PM

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Florent Foucaud, Hervé Hocquard, Dimitri Lajou. Complexity and algorithms for injective edge-coloring in graphs. Information Processing Letters, Elsevier, 2021, 170, pp.106121. ⟨10.1016/j.ipl.2021.106121⟩. ⟨hal-03201544⟩

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