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

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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.

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References

  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.

    Article  MathSciNet  Google Scholar 

  2. G. Cairns and Y. Nikolayevsky: Bounds for generalized thrackles, Discrete Comput. Geam.23(2) (2000), 191–206.

    Article  MathSciNet  Google Scholar 

  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.

    Article  MathSciNet  Google Scholar 

  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.

    MathSciNet  Google Scholar 

  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. J. F. Geelen, R. B. Richter and G. Salazar: Embedding grids in surfaces, European J. Combin.25(6) (2004), 785–792.

    Article  MathSciNet  Google Scholar 

  7. H. Hanani: Über wesentlich unplättbare Kurven im drei-dimensionalen Raume, Fundamenta Mathematicae23 (1934), 135–142.

    Article  Google Scholar 

  8. B. Mohar: Combinatorial local planarity and the width of graph embeddings, Canad. J. Math.44(6) (1992), 1272–1288.

    Article  MathSciNet  Google Scholar 

  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.

    MATH  Google Scholar 

  10. J. Pach and G. Tóth: Which crossing number is it anyway?, J. Combin. Theory Ser. B80(2) (2000), 225–246.

    Article  MathSciNet  Google Scholar 

  11. M. J. Pelsmajer, M. Schaefer and D. Stasi: Strong Hanani-Tutte on the projective plane. SiamJ. Discrete Math.23(3) (2009), 1317–1323.

    Article  MathSciNet  Google Scholar 

  12. M. J. Pelsmajer, M. Schaefer and D. štefankovič: Removing even crossings, J. Combin. Theory Ser. B97(4) (2007), 480–500.

    Article  MathSciNet  Google Scholar 

  13. M. J. Pelsmajer, M. Schaefer and D. štefankovič: Removing even crossings on surfaces. European J. Combin.30(7) (2009), 1704–1717.

    Article  MathSciNet  Google Scholar 

  14. N. Robertson and P. D. Seymour: Graph minors. VIII. A Kuratowski theorem for general surfaces, J. Combin. Theory Ser. B48(2) (1990), 255–288.

    Article  MathSciNet  Google Scholar 

  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. M. Schaefer: The graph crossing number and its variants: A survey, Electron. J. Combin., Dynamic Survey 21 (2017).

    Google Scholar 

  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.

    Chapter  Google Scholar 

  18. C. Thomassen: A simpler proof of the excluded minor theorem for higher surfaces, J. Combin. Theory Ser. B70(2) (1997), 306–311.

    Article  MathSciNet  Google Scholar 

  19. W. T. Tutte: Toward a theory of crossing numbers, J. Combinatorial Theory8 (1970), 45–53.

    Article  MathSciNet  Google Scholar 

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Correspondence to Radoslav Fulek.

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Supported by the People Programme (Marie Curie Actions) of the European Union’s Seventh Framework Programme (FP7/2007-2013) under REA grant agreement no [291734] and by Austrian Science Fund (FWF): M2281-N35.

Supported by project 16-01602Y of the Czech Science Foundation (GAČR) and by Charles University project UNCE/SCI/004.

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Fulek, R., Kynčl, J. Counterexample to an Extension of the Hanani-Tutte Theorem on the Surface of Genus 4. Combinatorica 39, 1267–1279 (2019). https://doi.org/10.1007/s00493-019-3905-7

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  • DOI: https://doi.org/10.1007/s00493-019-3905-7

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