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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01.09.2021
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| Abstract | •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. |
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| AbstractList | •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. 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. |
| ArticleNumber | 106121 |
| Author | Hocquard, Hervé Foucaud, Florent Lajou, Dimitri |
| Author_xml | – sequence: 1 givenname: Florent surname: Foucaud fullname: Foucaud, Florent email: florent.foucaud@uca.fr organization: LIMOS, CNRS UMR 6158, Université Clermont Auvergne, Aubière, France – sequence: 2 givenname: Hervé surname: Hocquard fullname: Hocquard, Hervé organization: Univ. Bordeaux, Bordeaux INP, CNRS, LaBRI, UMR5800, F-33400 Talence, France – sequence: 3 givenname: Dimitri surname: Lajou fullname: Lajou, Dimitri organization: Univ. Bordeaux, Bordeaux INP, CNRS, LaBRI, UMR5800, F-33400 Talence, France |
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| Cites_doi | 10.1016/S0012-365X(01)00466-6 10.1016/j.disc.2018.10.018 10.1142/S1793830918500222 10.1007/s00453-007-9044-3 10.1016/0196-6774(83)90032-9 10.2298/FIL1919411C 10.1016/0304-3975(76)90059-1 10.1016/j.ipl.2013.07.026 10.1016/j.jctb.2004.11.001 10.1137/0210055 |
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| Keywords | Subcubic graphs Graph algorithms Injective edge-coloring Planar graphs Treewidth |
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| SubjectTerms | Computer Science Data Structures and Algorithms Graph algorithms Injective edge-coloring Planar graphs Subcubic graphs Treewidth |
| Title | Complexity and algorithms for injective edge-coloring in graphs |
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