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Planar Lattice Graphs with Gallai’s Property

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Abstract

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.

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References

  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. Grünbaum B.: Vertices missed by longest paths or circuits. J. Comb. Theory A. 17, 31–38 (1974)

    Article  MATH  Google Scholar 

  3. Menke B.: Longest cycles in grid graphs. Studia Sci. Math. Hung. 36, 201–230 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  4. Menke B., Zamfirescu Ch., Zamfirescu T.: Intersection of longest cycles in grid graphs. J. Graph Theory 25, 37–52 (1997)

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  Google Scholar 

  6. Walther, H.: Über die Nichtexistenz eines Knotenpunktes, durch den alle längstenWege eines Graphen gehen. J. Comb. Theory 6, 1–6 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  7. Walther, H., Voss, H.-J.: Über Kreise in Graphen. VEB Deutscher Verlag der Wissenschaften, Berlin (1974)

  8. Zamfirescu C.T., Zamfirescu T.I.: A planar hypohamiltonian graph with 48 vertices. J. Graph Theory 55(4), 338–342 (2007)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Abdus Salam School of Mathematical Sciences, GC University, 68-B, New Muslim Town, Lahore, Pakistan

    Faisal Nadeem, Ayesha Shabbir & Tudor Zamfirescu

  2. Faculty of Mathematics, University of Dortmund, 44221, Dortmund, Germany

    Tudor Zamfirescu

  3. Institute of Mathematics, Roumanian Academy, Bucharest, Roumania

    Tudor Zamfirescu

Authors
  1. Faisal Nadeem
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  2. Ayesha Shabbir
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  3. Tudor Zamfirescu
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Correspondence to Tudor Zamfirescu.

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Nadeem, F., Shabbir, A. & Zamfirescu, T. Planar Lattice Graphs with Gallai’s Property. Graphs and Combinatorics 29, 1523–1529 (2013). https://doi.org/10.1007/s00373-012-1177-8

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  • DOI: https://doi.org/10.1007/s00373-012-1177-8

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