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Caterpillars are Antimagic

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Abstract

An antimagic labeling of a graph G is a bijection from the set of edges E(G) to \(\{1,2,\ldots ,|E(G)|\}\), such that all vertex sums are pairwise distinct, where the vertex sum at vertex u is the sum of the labels assigned to the edges incident to u. A graph is called antimagic when it has an antimagic labeling. Hartsfield and Ringel conjectured that every simple connected graph other than \(K_2\) is antimagic and the conjecture remains open even for trees. Here, we prove that caterpillars are antimagic by means of an \(O(n \log n)\) algorithm.

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Acknowledgements

Antoni Lozano is supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement ERC-2014-CoG 648276 AUTAR). Mercè Mora is supported by projects Gen. Cat. DGR 2017SGR1336, MINECO MTM2015-63791-R, and H2020-MSCA-RISE project 734922-CONNECT. Carlos Seara is supported by projects Gen. Cat. DGR 2017SGR1640, MINECO MTM2015-63791-R, and H2020-MSCA-RISE project 734922-CONNECT. Joaquín Tey is supported by project PRODEP-12612731.

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Authors and Affiliations

  1. Computer Science Department, Universitat Politècnica de Catalunya, Barcelona, Spain

    Antoni Lozano

  2. Mathematics Department, Universitat Politècnica de Catalunya, Barcelona, Spain

    Mercè Mora & Carlos Seara

  3. Mathematics Department, Universidad Autóonoma Metropolitana-Iztapalapa, Mexico City, Mexico

    Joaquín Tey

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  1. Antoni Lozano
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  2. Mercè Mora
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  3. Carlos Seara
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  4. Joaquín Tey
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Correspondence to Carlos Seara.

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This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska–Curie Grant agreement no. 734922.

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Lozano, A., Mora, M., Seara, C. et al. Caterpillars are Antimagic. Mediterr. J. Math. 18, 39 (2021). https://doi.org/10.1007/s00009-020-01688-z

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  • DOI: https://doi.org/10.1007/s00009-020-01688-z

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