Graphs and Combinatorics

, Volume 29, Issue 5, pp 1523–1529 | Cite as

Planar Lattice Graphs with Gallai’s Property

  • Faisal Nadeem
  • Ayesha Shabbir
  • Tudor Zamfirescu
Original Paper


In 1966 T. Gallai asked whether connected graphs with empty intersection of their longest paths do or do not exist. After examples of such graphs were found, the question was extended to graphs of higher connectivity, and to cycles instead of paths. Examples being again found, for connectivity up to 3, the question has been asked whether there exist large families of graphs without Gallai’s property. The family of grid graphs, a special kind of graphs embedded in the planar square lattice \({\mathcal {L}}\) , has been shown by B. Menke to contain no graph enjoying Gallai’s property. In this paper we find several examples of graphs embedded in \({\mathcal {L}}\) and enjoying that property with respect to both paths and cycles.


Lattice graphs Longest paths Longest cycles 


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  1. 1.
    Gallai T.: Problem 4. In: Erdös, P., Katona, G. (eds) Theory of Graphs, Proc Tihany 1966., p. 362. Academic Press, New York (1968)Google Scholar
  2. 2.
    Grünbaum B.: Vertices missed by longest paths or circuits. J. Comb. Theory A. 17, 31–38 (1974)zbMATHCrossRefGoogle Scholar
  3. 3.
    Menke B.: Longest cycles in grid graphs. Studia Sci. Math. Hung. 36, 201–230 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Menke B., Zamfirescu Ch., Zamfirescu T.: Intersection of longest cycles in grid graphs. J. Graph Theory 25, 37–52 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Schmitz, W.: Über längste Wege und Kreise in Graphen. Rend. Sem. Mat. Univ. Padova 53, 97–103 (1975)MathSciNetGoogle Scholar
  6. 6.
    Walther, H.: Über die Nichtexistenz eines Knotenpunktes, durch den alle längstenWege eines Graphen gehen. J. Comb. Theory 6, 1–6 (1969)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Walther, H., Voss, H.-J.: Über Kreise in Graphen. VEB Deutscher Verlag der Wissenschaften, Berlin (1974)Google Scholar
  8. 8.
    Zamfirescu C.T., Zamfirescu T.I.: A planar hypohamiltonian graph with 48 vertices. J. Graph Theory 55(4), 338–342 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Zamfirescu T.: A two-connected planar graph without concurrent longest paths. J. Combin. Theory B 13, 116–121 (1972)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Zamfirescu T.: On longest paths and circuits in graphs. Math. Scand. 38, 211–239 (1976)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Zamfirescu T.: Intersecting longest paths or cycles: a short survey. Analele Univ. Craiova. Seria Mat. Inf. 28, 1–9 (2001)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer 2012

Authors and Affiliations

  • Faisal Nadeem
    • 1
  • Ayesha Shabbir
    • 1
  • Tudor Zamfirescu
    • 1
    • 2
    • 3
  1. 1.Abdus Salam School of Mathematical SciencesGC UniversityLahorePakistan
  2. 2.Faculty of MathematicsUniversity of DortmundDortmundGermany
  3. 3.Institute of MathematicsRoumanian AcademyBucharestRoumania

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