Graphs and Combinatorics

, Volume 29, Issue 4, pp 729–733 | Cite as

An Infinite Family of Planar Hypohamiltonian Oriented Graphs

  • Susan A. van Aardt
  • Alewyn P. Burger
  • Marietjie Frick
Original Paper

Abstract

Carsten Thomassen asked in 1976 whether there exists a planar hypohamiltonian oriented graph. We answer his question by presenting an infinite family of planar hypohamiltonian oriented graphs, the smallest of which has order 9. A computer search showed that 9 is the smallest possible order of a hypohamiltonian oriented graph.

Keywords

Hypohamiltonian Bihomogeneously traceable Oriented graph Planar graph 

Mathematics Subject Classification

05C20 05C38 

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References

  1. 1.
    Hahn S., Zamfirescu T.: Bihomogeneously traceable oriented graphs. Rend. Sem. Mat Univ. Politec. Torino 39(2), 137–145 (1981)MathSciNetMATHGoogle Scholar
  2. 2.
    Skupień, Z.: On homogeneously traceable nonhamiltonian digraphs and oriented graphs. In: Chartrand, G., Alavi, Y., Goldsmith, D.L., Lesniak-Foster, L., Lick, D.R. (eds.) The Theory and Applications of Graphs (Proc. Fourth Int. Conf. on T.A.G., held in Kalamazoo, MI., 1980), pp. 517–527. Wiley (1981)Google Scholar
  3. 3.
    Thomassen, C.: Hypohamiltonian graphs and digraphs. In: Proceedings of the International Conference on the Theory and Applications of Graphs, Kalamazoo, 1976, pp. 557–571. Springer Verlag, Berlin (1978)Google Scholar
  4. 4.
    van Aardt S.A., Frick M., Katrenič P., Nielsen M.H.: The order of hypotraceable oriented graphs. Discret. Math. 311, 1273–1280 (2011)MATHCrossRefGoogle Scholar
  5. 5.
    Zamfirescu C.T.: An infinite family of planar non-hamiltonian bihomogeneusly traceable oriented graphs. Graphs Comb. 26, 141–146 (2010)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer 2012

Authors and Affiliations

  • Susan A. van Aardt
    • 1
  • Alewyn P. Burger
    • 2
  • Marietjie Frick
    • 1
  1. 1.Department of Mathematical SciencesUniversity of South Africa (UNISA)PretoriaSouth Africa
  2. 2.Department of LogisticsUniversity of StellenboschStellenboschSouth Africa

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