Skip to main content
SpringerLink
Log in

The number of rooted essential maps on surfaces

  • Published:
Acta Mathematica Sinica, English Series Aims and scope Submit manuscript

Abstract

In this paper, general functional equations of rooted essential maps on surfaces (orientable and nonorientable) are deduced and their formal solutions are presented. Further, three explicit formulae for counting essential maps on S 2, \( \widetilde{N_3 } \) and \( \widetilde{N_4 } \) are given. In the same time, some known results can be derived.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Brown, W. G.: On the enumeration of non-planar maps, Mem. Amer. Math. Soc., 65, (1966)

  2. Walsh, T., Lehman, A. B.: Counting rooted maps by genus I. J. Comb. Theory, Series B, 13, 192–218 (1972)

    Article  MATH  MathSciNet  Google Scholar 

  3. Walsh, T., Lehman, A. B.: Counting rooted maps by genus II. J. Comb. Theory, Series B, 13, 122–141 (1972)

    Article  MathSciNet  Google Scholar 

  4. Walsh, T., Lehman, A. B.: Counting rooted maps by genus III. J. Comb. Theory, Series B, 18, 222–259 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  5. Arquès, D.: Relation fonctionelles et dénombremant des cartes pointées sur letore. J. Comb. Theory, Series B, 43, 253–274 (1987)

    Article  MATH  Google Scholar 

  6. Arquès, D., Giorgetti, A.: Counting rooted maps on a surface. Theoretical Computer Science, 234, 255–272 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  7. Bender, E. A., Canfield, E. R., Robinson, R. W.: The enumeration of maps on the torus and the projective plane. Canada. Math. Bull., 31, 257–271 (1988)

    MATH  MathSciNet  Google Scholar 

  8. Gao, Z. C.: The number of rooted 2-connected triangular maps on the pojective plane. J. Comb. Theory, Series B, 53, 130–142 (1991)

    Article  MATH  Google Scholar 

  9. Liu, Y. P.: Enumerative Theory of Maps, Kluwer, Dordrecht/Boston/London, 1999

    MATH  Google Scholar 

  10. Liu, Y. P.: Theory of Counting Combinatorial Maps (in Chinese), Science Press, Beijing, 2001

    Google Scholar 

  11. Liu, Y. P.: Advances in Combinatorial Maps in Chinese, Northern Jiaotong University Press, Beijing, 2003

    Google Scholar 

  12. Hao, R. X., Cai, J. L., Liu, Y. P.: Counting g-essential maps with small genera. J. Appl. Math. & Computing, 9(2), 451–463 (2002)

    MATH  MathSciNet  Google Scholar 

  13. Liu, W. Z., Liu, Y. P.: Enumeration of three kinds of rooted maps on the Klein bottle. J. Appl. Math. & Computing, 24(1–2), 411–419 (2007)

    Article  MATH  Google Scholar 

  14. Tutte, W. T.: A census of planar triangulations. Canad. J. Math, 14, 21–38 (1962)

    MATH  MathSciNet  Google Scholar 

  15. Tutte, W. T.: A census of slicings. Canad. J. Math, 14, 708–722 (1962)

    MATH  MathSciNet  Google Scholar 

  16. Tutte, W. T.: A census of planar maps. Canad. J. Math, 15, 249–271 (1963)

    MATH  MathSciNet  Google Scholar 

  17. Tutte, W. T.: The enumerative theory of planar maps, in A Survey of Combinatorial Theory, J. N. Srivatava et al., eds., North Holland, 437–448, 1973

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, P. R. China

    Wen Zhong Liu

  2. Department of Mathematics, Beijing Jiaotong University, Beijing, 100044, P. R. China

    Yan Pei Liu

Authors
  1. Wen Zhong Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yan Pei Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Wen Zhong Liu.

Additional information

Supported by NNSFC under Grant 603730308 and 10571013

Rights and permissions

Reprints and permissions

About this article

Cite this article

Liu, W.Z., Liu, Y.P. The number of rooted essential maps on surfaces. Acta. Math. Sin.-English Ser. 24, 1303–1312 (2008). https://doi.org/10.1007/s10114-008-6181-9

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10114-008-6181-9

Keywords

MR(2000) Subject Classification

Search

Navigation