Skip to main content

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


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.

This is a preview of subscription content, access via your institution.


  1. D.P. Agrawal, Graph theoretical analysis and designs of multistage interconnection networks. IEEE Trans. Comput. 32, 637–648 (1983)

  2. T. Gallai, Problem 4, in Theory of Graphs, Proc. Tihany 1966, ed. by P. Erdös, G. Katona ((Academic Press, New York, 1968), p. 362

    Google Scholar 

  3. J.P. Hayes, A graph model for fault-tolerant computing systems. IEEE Trans. Comput. 25, 875–884 (1976)

    MathSciNet  Article  MATH  Google Scholar 

  4. D.A. Holton, J. Sheehan, The Petersen Graph, Australian Math. Soc., Lecture Series (No. 7) (Cambridge University Press, Cambridge, 1993)

    Book  MATH  Google Scholar 

  5. S. Jendrol, Z. Skupień, Exact number of longest cycles with empty intersection. Eur. J. Comb. 18, 575–578 (1974)

    Article  MATH  Google Scholar 

  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. E. Máčajová, M. Škoviera, Infinitely many hypohamiltonian cubic graphs of girth 7. Graphs Comb. 27, 231–241 (2011)

    MathSciNet  Article  MATH  Google Scholar 

  8. F. Nadeem, A. Shabbir, T. Zamfirescu, Planar lattice graphs with Gallai’s property. Graphs Comb. 29, 1523–1529 (2013)

  9. W. Schmitz, Über längste Wege und Kreise in Graphen. Rend. Sem. Mat. Univ. Padova 53, 97–103 (1975)

    MathSciNet  MATH  Google Scholar 

  10. T. Zamfirescu, A two-connected planar graph without concurrent longest paths. J. Comb. Theory B 13, 116–121 (1972)

    MathSciNet  Article  MATH  Google Scholar 

  11. T. Zamfirescu, On longest paths and circuits in graphs. Math. Scand. 38, 211–239 (1976)

    MathSciNet  MATH  Google Scholar 

  12. T. Zamfirescu, Intersecting longest paths or cycles: a short survey. Analele Univ. Craiova. Seria Mat. Inf. 28, 1–9 (2001)

    MathSciNet  MATH  Google Scholar 

Download references


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.

Author information

Authors and Affiliations


Corresponding author

Correspondence to Tudor Zamfirescu.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Shabbir, A., Zamfirescu, T. Gallai’s property for graphs in lattices on the torus and the Möbius strip. Period Math Hung 72, 1–11 (2016).

Download citation

  • Published:

  • Issue Date:

  • DOI:


  • Longest paths
  • Longest cycles
  • Square lattice
  • Hexagonal lattice
  • Torus
  • Möbius strip