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Graphs and Combinatorics

, Volume 32, Issue 6, pp 2339–2353 | Cite as

Some Triangulated Surfaces without Balanced Splitting

  • Vincent Despré
  • Francis Lazarus
Original Paper

Abstract

Let G be the graph of a triangulated surface \(\varSigma \) of genus \(g\ge 2\). A cycle of G is splitting if it cuts \(\varSigma \) into two components, neither of which is homeomorphic to a disk. A splitting cycle has type k if the corresponding components have genera k and \(g-k\). It was conjectured that G contains a splitting cycle (Barnette 1982). We confirm this conjecture for an infinite family of triangulations by complete graphs but give counter-examples to a stronger conjecture (Mohar and Thomassen in Graphs on surfaces. Johns Hopkins studies in the mathematical sciences. Johns Hopkins University Press, Baltimore, 2001) claiming that G should contain splitting cycles of every possible type.

Keywords

Combinatorial surfaces Splitting cycles Complete graphs 

Mathematics Subject Classification

05C10 57M07 68R99 

Notes

Acknowledgments

The authors would like to thank Thom Sulanke for interesting discussions about his code for generating irreducible triangulations. We are also grateful to the anonymous reviewer for her/his detailed comments and suggestions.

References

  1. 1.
    Barnette, D.W., Edelson, A.L.: All orientable 2-manifolds have finitely many minimal triangulations. Israel J. Math. 62(1), 90–98 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Barnette, D.W., Edelson, A.L.: All 2-manifolds have finitely many minimal triangulations. Isr. J. Math. 67(1), 123–128 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Boulch, A., Colin de Verdière, É., Nakamoto, A.: Irreducible triangulations of surfaces with boundary. Graphs Comb. 29(6), 1675–1688 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Cabello, S., Colin de Verdière, É., Lazarus, F.: Finding cycles with topological properties in embedded graphs. SIAM J. Discret. Math. 25(4), 1600–1614 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Chambers, E., Colin de Verdière, E., Erickson, J., Lazarus, F., Whittlesey, K.: Splitting (complicated) surfaces is hard. In: Proc. 22nd annual Symp. Comput. Geom., ACM, pp. 421–429 (2006)Google Scholar
  6. 6.
    Ellingham, M.N., Stephens, C.: Triangular embeddings of complete graphs (neighborly maps) with 12 and 13 vertices. J. Comb. Des. 13(5), 336–344 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Ellingham, M.N., Zha, X.: Separating cycles in doubly toroidal embeddings. Graphs Comb. 19(2), 161–175 (2003)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Grannell, M.J., Knor, M.: On the number of triangular embeddings of complete graphs and complete tripartite graphs. J. Graph Theory 69(4), 370–382 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Gross, J.L., Tucker, T.W.: Topological Graph Theory. Wiley, Dover (1987) (reprint 2001 from)Google Scholar
  10. 10.
    Jennings, D.L.: Separating cycles in triangulations of the double torus. Ph.D. thesis, Vanderbilt University (2003)Google Scholar
  11. 11.
    Joret, G., Wood, D.R.: Irreducible triangulations are small. J. Comb. Theory, Ser. B 100(5), 446–455 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Korzhik, V.P., Voss, H.J.: On the number of nonisomorphic orientable regular embeddings of complete graphs. J. Comb. Theory, Ser. B 81(1), 58–76 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Lawrencenko, S., Negami, S., White, A.T.: Three nonisomorphic triangulations of an orientable surface with the same complete graph. Discret. Math. 135(1), 367–369 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Mohar, B., Thomassen, C.: Graphs on Surfaces. Johns Hopkins Studies in the Mathematical Sciences. Johns Hopkins University Press, Baltimore (2001)Google Scholar
  15. 15.
    Nakamoto, A., Ota, K.: Note on irreducible triangulations of surfaces. J. Graph Theory 20(2), 227–233 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Ringel, G.: Map Color Theorem, vol. 209. Springer, Berlin (1974)CrossRefzbMATHGoogle Scholar
  17. 17.
    Robertson, N., Thomas, R.: On the orientable genus of graphs embedded in the klein bottle. J. Graph Theory 15(4), 407–419 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Stillwell, J.: Classical Topology and Combinatorial Group Theory. Graduate Texts in Mathematics. Springer, Berlin (1993)CrossRefzbMATHGoogle Scholar
  19. 19.
    Sulanke, T.: Generating irreducible triangulations of surfaces (2006). arXiv:math/0606687
  20. 20.
    Sulanke, T.: Irreducible triangulations of low genus surfaces (2006). arXiv:math/0606690
  21. 21.
    Zha, X., Zhao, Y.: On non-null separating circuits in embedded graphs. Contemp. Math. 147, 349–362 (1993)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Japan 2016

Authors and Affiliations

  1. 1.Gipsa-labUniversité de Grenoble AlpesGrenobleFrance
  2. 2.Gipsa-labCNRSGrenobleFrance

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