Skip to main content
Log in

Hypohamiltonian Oriented Graphs of All Possible Orders

  • Original Paper
  • Published:
Graphs and Combinatorics Aims and scope Submit manuscript

Abstract

A digraph is hamiltonian if it has a cycle that visits every vertex. If a digraph \(D\) is nonhamiltonian and \(D-v\) is hamiltonian for every \(v\in V(D)\), then \(D\) is said to be hypohamiltonian. It is known that there exist hypohamiltonian digraphs of order \(n\) for every \(n\ge 6\). Several infinite families of hypohamiltonian oriented graphs have appeared in the literature, but there are infinitely many orders which are not covered by those constructions. In this paper we construct a hypohamiltonian oriented graph of order \(n\) for every \(n\ge 9\).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

References

  1. van Aardt, S.A., Burger, A.P., Frick, M.: An infinite family of planar hypohamiltonian oriented graphs. Gr. Comb. 29(4), 729–733 (2013)

    Article  MATH  Google Scholar 

  2. van Aardt, S.A., Burger, A.P., Frick, M., Llano, B., Zuazua, R.: Infinite families of 2-hypohamiltonian/2-hypohamiltonian oriented graphs. Gr. Comb. 30(4), 783–800 (2014)

    Article  MATH  Google Scholar 

  3. Aldred, R.E.L., McKay, B.D., Wormald, N.C.: Small hypohamiltonian graphs. J. Comb. Math. Comb. Comput. 23, 143–152 (1997)

    MathSciNet  MATH  Google Scholar 

  4. Bang-Jensen, J., Gutin, G.: Digraphs: Theory, Algorithms and Applications, 2nd edn. Springer, London (2009)

    Book  Google Scholar 

  5. Burger, A.P.: Computational results on the traceability of oriented graphs of small order. Electron. J. Comb. 20(4), #P23 (2013)

  6. Fouquet, J.-L., Jolivet, J.-L.: Graphes hypohamiltoniens orientes. In: Problémes Combinatoires et Théorie des Graphes (1976), Colloq. Int. CNRS No. 260, pp. 149–151 (1978)

  7. Grötschel, M., Wakabayashi, Y.: Hypohamiltonan digraphs. Math. Oper. Res. 36, 99–229 (1980)

    MATH  Google Scholar 

  8. Locke, S.C., Witte, D.: On non-hamiltonian circulant digraphs of outdegree three. J. Gr. Theory 30, 319–331 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  9. Penn, L.E., Witte, D.: When the cartesian product of two directed cycles is hypohamiltonian. J. Gr. Theory 7, 441–443 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  10. Rankin, R.A.: A campanological problem in group theory. Proc. Camb. Phil. Soc. 44, 17–25 (1948)

    Article  MathSciNet  MATH  Google Scholar 

  11. Skupień, Z.: Hypohamiltonian/hypotraceable digraphs abound. J. Comb. Math. Comb. Comput. 24, 239–242 (1997)

    MATH  Google Scholar 

  12. Thomassen, C.: Hypohamiltonian graphs and digraphs. In: Proceedings of the International Conference on the Theory and Applications of Graphs, Kalamazoo, 1976, Springer, pp. 557–571 (1978)

Download references

Acknowledgments

The authors wish to thank the University of South Africa for sponsoring the Salt Rock Workshop of 16–28 March 2014, which resulted in this paper.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Arnfried Kemnitz.

Additional information

This material is based upon work supported by the National Research Foundation of South Africa under Grant Numbers 77248, 81004 and 81075.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

van Aardt, S.A., Burger, A.P., Frick, M. et al. Hypohamiltonian Oriented Graphs of All Possible Orders. Graphs and Combinatorics 31, 1821–1831 (2015). https://doi.org/10.1007/s00373-015-1561-2

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00373-015-1561-2

Keywords

Navigation