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Order of the smallest counterexample to Gallai’s conjecture

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Abstract

In 1966, Gallai conjectured that all the longest paths of a connected graph have a common vertex. Zamfirescu conjectured that the smallest counterexample to Gallai’s conjecture is a graph on 12 vertices. We prove that Gallai’s conjecture is true for every connected graph G with α′(G) ≤ 5, which implies that Zamfirescu’s conjecture is true.

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References

  1. P. Balister, E. Györi, J. Lehel, R. Schelp: Longest paths in circular arc graphs. Comb. Probab. Comput. 13 (2004), 311–317.

    Article  MathSciNet  MATH  Google Scholar 

  2. G. Brinkmann, N. Van Cleemput: Private communication.

  3. F. Chen: Nonempty intersection of longest paths in a graph with a small matching number. Czech. Math. J. 65 (2015), 545–553.

    Article  MathSciNet  MATH  Google Scholar 

  4. G. Chen, J. Ehrenmüller, C. G. Fernandes, C. G. Heise, S. Shan, P. Yang, A. N. Yates: Nonempty intersection of longest paths in series-parallel graphs. Discrete Math. 340 (2017), 287–304.

    Article  MathSciNet  MATH  Google Scholar 

  5. S. F. de Rezende, C. G. Fernandes, D. M. Martin, Y. Wakabayashi: Intersecting longest paths. Discrete Math. 313 (2013), 1401–1408.

    Article  MathSciNet  MATH  Google Scholar 

  6. T. Gallai: Problem 4. Theory of Graphs. Proceedings of the colloquium held at Tihany, Hungary, September 1966 (P. Erdös, G. Katona, eds.). Academic Press, New York, 1968.

    Google Scholar 

  7. S. Klavžar, M. Petkovšek: Graphs with nonempty intersection of longest paths. Ars Comb. 29 (1990), 43–52.

    MathSciNet  MATH  Google Scholar 

  8. J. Petersen: Die Theorie der regulären graphs. Acta Math. 15 (1891), 193–220. (In German.)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

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

    MATH  Google Scholar 

  11. T. I. Zamfirescu: L’histoire et l’etat présent des bornes connues pour \(P_k^j,C_k^j,\bar P_k^jet\bar C_k^j\) . Cah. Cent. Étud. Rech. Opér. 17 (1975), 427–439. (In French.)

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Anhui University of Finance and Economics, No. 962 Caoshan Road, Bengbu City, P.R. China

    Fuyuan Chen

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  1. Fuyuan Chen
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Correspondence to Fuyuan Chen.

Additional information

This research was supported by NSFC Grant 11601001.

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Chen, F. Order of the smallest counterexample to Gallai’s conjecture. Czech Math J 68, 341–369 (2018). https://doi.org/10.21136/CMJ.2018.0422-16

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  • DOI: https://doi.org/10.21136/CMJ.2018.0422-16

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