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Infinite families of 2-hypohamiltonian/2-hypotraceable oriented graphs

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Abstract

A digraph D of order n is r-hypohamiltonian (respectively r-hypotraceable) for some positive integer r < n − 1 if D is nonhamiltonian (nontraceable) and the deletion of any r of its vertices leaves a hamiltonian (traceable) digraph. A 1-hypohamiltonian (1-traceable) digraph is simply called hypohamiltonian (hypotraceable). Although hypohamiltonian and hypotraceable digraphs are well-known and well-studied concepts, we have found no mention of r-hypohamiltonian or r-hypotraceable digraphs in the literature for any r > 1. In this paper we present infinitely many 2-hypohamiltonian oriented graphs and use these to construct infinitely many 2-hypotraceable oriented graphs. We also discuss an interesting connection between the existence of r-hypotraceable oriented graphs and the Path Partition Conjecture for oriented graphs.

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Authors and Affiliations

  1. Department of Mathematical Sciences, University of South Africa (UNISA), PO BOX 392, Pretoria, 0003, South Africa

    Susan A. van Aardt & Marietjie Frick

  2. Department of Logistics, University of Stellenbosch, Private Bag X1, Matieland, Stellenbosch, 7602, South Africa

    Alewyn P. Burger

  3. Departamento de Matemáticas, Universidad Autónoma Metropolitana-Iztapalapa, Av. San Rafael Atlixco 186, Colonia Vicentina, Mexico, DF, 09340, Mexico

    Bernardo Llano

  4. Departamento de Matemáticas, Universidad Nacional Autónoma de México, Ciudad Universitaria, Coyoacan, 04510, Mexico, DF, Mexico

    Rita Zuazua

Authors
  1. Susan A. van Aardt
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  2. Alewyn P. Burger
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  3. Marietjie Frick
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  4. Bernardo Llano
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  5. Rita Zuazua
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Corresponding author

Correspondence to Alewyn P. Burger.

Additional information

This material is based upon work supported by the National Research Foundation of South Africa under Grant numbers 77248 and 71308.

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van Aardt, S.A., Burger, A.P., Frick, M. et al. Infinite families of 2-hypohamiltonian/2-hypotraceable oriented graphs. Graphs and Combinatorics 30, 783–800 (2014). https://doi.org/10.1007/s00373-013-1312-1

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  • DOI: https://doi.org/10.1007/s00373-013-1312-1

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