Complexity and algorithms for injective edge-coloring in graphs

•We start a systematic study of the complexity of INJECTIVE k-EDGE-COLORING.•INJECTIVE k-EDGE-COLORING for k=3,4 is hard for restricted classes of subcubic graphs.•INJECTIVE k-EDGE-COLORING is linear-time solvable on graphs of bounded treewidth.•All planar bipartite subcubic graphs of girth 16 are i...

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Veröffentlicht in:Information processing letters Jg. 170; S. 106121
Hauptverfasser: Foucaud, Florent, Hocquard, Hervé, Lajou, Dimitri
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Elsevier B.V 01.09.2021
Elsevier
Schlagworte:
Computer Science
Data Structures and Algorithms
Graph algorithms
Injective edge-coloring
Planar graphs
Subcubic graphs
Treewidth
Subcubic graphs
Graph algorithms
Injective edge-coloring
Planar graphs
Treewidth
ISSN:0020-0190, 1872-6119
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Zusammenfassung:•We start a systematic study of the complexity of INJECTIVE k-EDGE-COLORING.•INJECTIVE k-EDGE-COLORING for k=3,4 is hard for restricted classes of subcubic graphs.•INJECTIVE k-EDGE-COLORING is linear-time solvable on graphs of bounded treewidth.•All planar bipartite subcubic graphs of girth 16 are injectively 3-edge-colorable. 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 Injectivek-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 Injectivek-Edge-Coloring remains NP-complete even for graphs of maximum degree at most 53k. In contrast with these negative results, we show that Injectivek-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.
ISSN:0020-0190
1872-6119
DOI:10.1016/j.ipl.2021.106121