Graphs and Combinatorics

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

Some Triangulated Surfaces without Balanced Splitting

Original Paper


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.


Combinatorial surfaces Splitting cycles Complete graphs 

Mathematics Subject Classification

05C10 57M07 68R99 


  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)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Boulch, A., Colin de Verdière, É., Nakamoto, A.: Irreducible triangulations of surfaces with boundary. Graphs Comb. 29(6), 1675–1688 (2013)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Ellingham, M.N., Zha, X.: Separating cycles in doubly toroidal embeddings. Graphs Comb. 19(2), 161–175 (2003)MathSciNetMATHGoogle 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)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Ringel, G.: Map Color Theorem, vol. 209. Springer, Berlin (1974)CrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Stillwell, J.: Classical Topology and Combinatorial Group Theory. Graduate Texts in Mathematics. Springer, Berlin (1993)CrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Japan 2016

Authors and Affiliations

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

Personalised recommendations