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Improved Sufficient Conditions for the Existence of Anti-Directed Hamiltonian Cycles in Digraphs

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Abstract

Let D be a directed graph of order n. An anti-directed (hamiltonian) cycle H in D is a (hamiltonian) cycle in the graph underlying D such that no pair of consecutive arcs in H form a directed path in D. In this paper we give sufficient conditions for the existence of anti-directed hamiltonian cycles. Specifically, we prove that a directed graph D of even order n with minimum indegree and outdegree greater than \({\frac{1}{2}n + 7\sqrt{n}/3}\) contains an anti-directed hamiltonian cycle. In addition, we show that D contains anti-directed cycles of all possible (even) lengths when n is sufficiently large and has minimum in- and out-degree at least \({(1/2+ \epsilon)n}\) for any \({\epsilon > 0}\) .

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Authors and Affiliations

  1. Department of Mathematics, University of Dayton, Dayton, USA

    Arthur H. Busch

  2. Department of Mathematical and Statistical Sciences, University of Colorado Denver, Denver, USA

    Michael S. Jacobson & Timothy Morris

  3. Department of Mathematics, Illinois State University, Normal, USA

    Michael J. Plantholt & Shailesh K. Tipnis

Authors
  1. Arthur H. Busch
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  2. Michael S. Jacobson
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  3. Timothy Morris
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  4. Michael J. Plantholt
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  5. Shailesh K. Tipnis
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Correspondence to Michael S. Jacobson.

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Busch, A.H., Jacobson, M.S., Morris, T. et al. Improved Sufficient Conditions for the Existence of Anti-Directed Hamiltonian Cycles in Digraphs. Graphs and Combinatorics 29, 359–364 (2013). https://doi.org/10.1007/s00373-011-1116-0

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  • DOI: https://doi.org/10.1007/s00373-011-1116-0

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