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Counterexample to an Extension of the Hanani-Tutte Theorem on the Surface of Genus 4

  • Radoslav FulekEmail author
  • Jan Kynčl
Article
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Abstract

We find a graph of genus 5 and its drawing on the orientable surface of genus 4 with every pair of independent edges crossing an even number of times. This shows that the strong Hanani–Tutte theorem cannot be extended to the orientable surface of genus 4. As a base step in the construction we use a counterexample to an extension of the unified Hanani–Tutte theorem on the torus.

Mathematics Subject Classification (2010)

05C10 57M15 

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References

  1. [1]
    J. Battle, F. Harary, Y. Kodama and J. W. T. Youngs: Additivity of the genus of a graph, Bull. Amer. Math. Soc. 68 (1962), 565–568.MathSciNetCrossRefGoogle Scholar
  2. [2]
    G. Cairns and Y. Nikolayevsky: Bounds for generalized thrackles, Discrete Comput. Geam. 23(2) (2000), 191–206.MathSciNetCrossRefGoogle Scholar
  3. [3]
    É. Colin de Verdière, V. Kaluža, P. Paták, Z. Patáková and M. Tancer: A direct proof of the strong Hanani-Tutte theorem on the projective plane, J. Graph Algorithms Appl. 21(5) (2017), 939–981.MathSciNetCrossRefGoogle Scholar
  4. [4]
    R. Fulek and J. Kynčl: The ℤ2-genus of Kuratowski minors, Proceedings of the 34th International Symposium on Computational Geometry (SoCG 2018), Leibniz International Proceedings in Informatics (LIPIcs) 99, 40:1–10:14, Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik. 2018.MathSciNetGoogle Scholar
  5. [5]
    R. Fulek, J. Kynčl and D. Pálvölgyi: Unified Hanani-Tutte theorem. Electron. J. Combin. 24(3) (2017), P3.18, 8 pp.Google Scholar
  6. [6]
    J. F. Geelen, R. B. Richter and G. Salazar: Embedding grids in surfaces, European J. Combin. 25(6) (2004), 785–792.MathSciNetCrossRefGoogle Scholar
  7. [7]
    H. Hanani: Über wesentlich unplättbare Kurven im drei-dimensionalen Raume, Fundamenta Mathematicae 23 (1934), 135–142.CrossRefGoogle Scholar
  8. [8]
    B. Mohar: Combinatorial local planarity and the width of graph embeddings, Canad. J. Math. 44(6) (1992), 1272–1288.MathSciNetCrossRefGoogle Scholar
  9. [9]
    B. Mohar and C. Thomassen: Graphs on surfaces, Johns Hopkins Studies in the Mathematical Sciences, Johns Hopkins University Press, Baltimore, MD (2001), ISBN 0-8018-6689-8.zbMATHGoogle Scholar
  10. [10]
    J. Pach and G. Tóth: Which crossing number is it anyway?, J. Combin. Theory Ser. B 80(2) (2000), 225–246.MathSciNetCrossRefGoogle Scholar
  11. [11]
    M. J. Pelsmajer, M. Schaefer and D. Stasi: Strong Hanani-Tutte on the projective plane. Siam J. Discrete Math. 23(3) (2009), 1317–1323.MathSciNetCrossRefGoogle Scholar
  12. [12]
    M. J. Pelsmajer, M. Schaefer and D. štefankovič: Removing even crossings, J. Combin. Theory Ser. B 97(4) (2007), 480–500.MathSciNetCrossRefGoogle Scholar
  13. [13]
    M. J. Pelsmajer, M. Schaefer and D. štefankovič: Removing even crossings on surfaces. European J. Combin. 30(7) (2009), 1704–1717.MathSciNetCrossRefGoogle Scholar
  14. [14]
    N. Robertson and P. D. Seymour: Graph minors. VIII. A Kuratowski theorem for general surfaces, J. Combin. Theory Ser. B 48(2) (1990), 255–288.MathSciNetCrossRefGoogle Scholar
  15. [15]
    M. Schaefer: Hanani-Tutte and related results, Geometry–Intuitive, Discrete, and Convex, vol. 24 of Bolyai Soc. Math. Stud., 259–299, János Bolyai Math. Soc, Budapest (2013).Google Scholar
  16. [16]
    M. Schaefer: The graph crossing number and its variants: A survey, Electron. J. Combin., Dynamic Survey 21 (2017).Google Scholar
  17. [17]
    M. Schaefer and D. štefankovič: Block additivity of ℤ2-embeddings, in: (Wismath S., Wolff A. eds) Graph Drawing. GD 2013. Lecture Notes in Computer Science, vol 8242. Springer, Cham, 2013.CrossRefGoogle Scholar
  18. [18]
    C. Thomassen: A simpler proof of the excluded minor theorem for higher surfaces, J. Combin. Theory Ser. B 70(2) (1997), 306–311.MathSciNetCrossRefGoogle Scholar
  19. [19]
    W. T. Tutte: Toward a theory of crossing numbers, J. Combinatorial Theory 8 (1970), 45–53.MathSciNetCrossRefGoogle Scholar

Copyright information

© János Bolyai Mathematical Society and Springer-Verlag 2019

Authors and Affiliations

  1. 1.IST AustriaKlosterneuburgAustria
  2. 2.Charles UniversityPragueCzech Republic

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