Exceptional Balanced Triangulations on Surfaces

Abstract

Izmestiev, Klee and Novik proved that any two balanced triangulations of a closed surface \(F^2\) can be transformed into each other by a sequence of six operations called basic cross flips. Recently Murai and Suzuki proved that among these six operations only two operations are almost sufficient in the sense that, with for finitely many exceptions, any two balanced triangulations of a closed surface \(F^2\) can be transformed into each other by these two operations. We investigate such finitely many exceptions, called exceptional balanced triangulations, and obtain the list of exceptional balanced triangulations of closed surfaces with low genera. Furthermore, we discuss the subsets \(\mathcal{O}\) of the six operations satisfying the property that any two balanced triangulations of the same closed surface can be connected through a sequence of operations from \(\mathcal{O}\).

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

References

  1. 1.

    Ellingham, M.N., Stephens, C., Zha, X.: The nonorientable genus of complete tripartite graphs. J. Combin. Theory, Ser. B 96, 529–559 (2006)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Izmestiev, I., Klee, S., Novik, I.: Simplicial moves on balanced complexes. Adv. Math. 320, 82–114 (2017)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Kawarabayashi, K., Nakamoto, A., Suzuki, Y.: \(N\)-Flips in even triangulations on surfaces. J. Combin. Theory, Ser. B. 99, 229–246 (2009)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Juhnke-Kubitzke, M., Venturello, L.: Balanced shellings and moveson balanced manifolds. arXiv:1804.06270

  5. 5.

    Murai, S., Suzuki, Y.: Balanced subdivisions and flips on surfaces. Proc. Am. Math. Soc. 146, 939–951 (2018)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Nakamoto, A., Sakuma, T., Suzuki, Y.: \(N\)-Flips in even triangulations on the sphere. J. Graph Theory 51, 260–268 (2006)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Pachner, U.: Konstruktionsmethoden und das kombinatorische Homöomorphieproblem für Triangulationen kompakter smilinearer Mannigfaltigkeiten. Abh. Math. Sem. Univ. Hamburg 57, 69–86 (1987)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Pachner, U.: P.L. homeomorphic manifolds are equivalent by elementary shellings. Eur. J. Combin. 12, 129–145 (1991)

    MathSciNet  Article  Google Scholar 

  9. 9.

    White, A.T.: The genus of the complete tripartite graph \(K_{mn, n, n}\). J. Combin. Theory 7, 283–285 (1969)

    Article  Google Scholar 

Download references

Acknowledgements

S. Klee: Research supported by NSF Grant DMS-1600048. S. Murai: Research supported by KAKENHI16K05102. Y. Suzuki: Research supported by KAKENHI16K05250.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Yusuke Suzuki.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Klee, S., Murai, S. & Suzuki, Y. Exceptional Balanced Triangulations on Surfaces. Graphs and Combinatorics 35, 1361–1373 (2019). https://doi.org/10.1007/s00373-018-2001-x

Download citation

Keywords

  • Balanced triangulation
  • Closed surface
  • Local transformation