Periodica Mathematica Hungarica

, Volume 72, Issue 1, pp 1–11 | Cite as

Gallai’s property for graphs in lattices on the torus and the Möbius strip

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Abstract

We prove the existence of graphs with empty intersection of their longest paths or cycles as subgraphs of lattices on the torus and the Möbius strip.

Keywords

Longest paths Longest cycles Square lattice Hexagonal lattice Torus Möbius strip 

Notes

Acknowledgments

The second author’s work was supported by a grant of the Roumanian National Authority for Scientific Research, CNCS—UEFISCDI, project number PN-II-ID-PCE-2011-3-0533.

References

  1. 1.
    D.P. Agrawal, Graph theoretical analysis and designs of multistage interconnection networks. IEEE Trans. Comput. 32, 637–648 (1983)Google Scholar
  2. 2.
    T. Gallai, Problem 4, in Theory of Graphs, Proc. Tihany 1966, ed. by P. Erdös, G. Katona ((Academic Press, New York, 1968), p. 362Google Scholar
  3. 3.
    J.P. Hayes, A graph model for fault-tolerant computing systems. IEEE Trans. Comput. 25, 875–884 (1976)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    D.A. Holton, J. Sheehan, The Petersen Graph, Australian Math. Soc., Lecture Series (No. 7) (Cambridge University Press, Cambridge, 1993)CrossRefMATHGoogle Scholar
  5. 5.
    S. Jendrol, Z. Skupień, Exact number of longest cycles with empty intersection. Eur. J. Comb. 18, 575–578 (1974)CrossRefMATHGoogle Scholar
  6. 6.
    M. Jooyandeh, B. D. Mckey, P. R. Östergård, V. H. Petersson, C. T. Zamfirescu, Planar hypohamiltonian graphs on 40 vertices, arXiv:1302.2698 [math.CO]
  7. 7.
    E. Máčajová, M. Škoviera, Infinitely many hypohamiltonian cubic graphs of girth 7. Graphs Comb. 27, 231–241 (2011)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    F. Nadeem, A. Shabbir, T. Zamfirescu, Planar lattice graphs with Gallai’s property. Graphs Comb. 29, 1523–1529 (2013)Google Scholar
  9. 9.
    W. Schmitz, Über längste Wege und Kreise in Graphen. Rend. Sem. Mat. Univ. Padova 53, 97–103 (1975)MathSciNetMATHGoogle Scholar
  10. 10.
    T. Zamfirescu, A two-connected planar graph without concurrent longest paths. J. Comb. Theory B 13, 116–121 (1972)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    T. Zamfirescu, On longest paths and circuits in graphs. Math. Scand. 38, 211–239 (1976)MathSciNetMATHGoogle Scholar
  12. 12.
    T. Zamfirescu, Intersecting longest paths or cycles: a short survey. Analele Univ. Craiova. Seria Mat. Inf. 28, 1–9 (2001)MathSciNetMATHGoogle Scholar

Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2015

Authors and Affiliations

  1. 1.Abdus Salam School of Mathematical SciencesGC UniversityLahorePakistan
  2. 2.Faculty of MathematicsUniversity of DortmundDortmundGermany
  3. 3.“Simion Stoïlow” Institute of Mathematics Roumanian AcademyBucharestRoumania

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