Skip to main content
SpringerLink
Log in

Counting rooted 4-regular unicursal planar maps

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

Abstract

A map is 4-regular unicursal if all its vertices are 4-valent except two odd-valent vertices. This paper investigates the number of rooted 4-regular unicursal planar maps and presents some formulae for such maps with four parameters: the number of edges, the number of inner faces and the valencies of the two odd vertices.

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. Bousquet-Mélou, M., Schaeffer, G. Enumeration of planar constellations. Adv. Appl. Math., 24: 337–368 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  2. Brown, W.G. Enumeration of quadrangular dissections of the disc. Canad. J. Math., 17: 302–317 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  3. Cai, J.L., Liu, Y.P. The number of nearly 4-regular planar maps. Utilitas Mathematica, 58: 243–254 (2000)

    MathSciNet  MATH  Google Scholar 

  4. Cai, J.L, Liu, Y.P. Enumerating rooted Eulerian planar maps. Acta Math. Sci., 21: 289–294 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  5. Cai, J.L. The number of rooted loopless eulerian planar maps. Acta Math. Appl. Sin., 29: 210–216 (2006) (in Chinese)

    MathSciNet  Google Scholar 

  6. Cai, J.L., Liu, Y.P. The number of rooted eulerian planar maps. Science in China Series A: Mathematics, 51: 2005–2012 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  7. Good, I.J. Generalizations to several varibles of Lagrange’s expansion, with applications to stochastic processes. Proc. Camb. Phil. Soc., 56: 367–380 (1960)

    Article  MATH  Google Scholar 

  8. Koganov, L.M., Liskovets, V.A., Walsh, T.R.S. Total vertex enumeration in rooted planar maps. Ars Combin., 54: 149–160 (2000)

    MathSciNet  MATH  Google Scholar 

  9. Liskovets, V.A. Enumeration of nonisomorphic planar maps. Selecta Math. Soviet, 4: 304–323 (1985)

    MATH  Google Scholar 

  10. Liskovets, V.A. A pattern of asymptotic vertex valency distributions in planar maps. J. Combin. Theory Ser. B, 75: 116–133 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  11. Liskovets, V.A., Walsh, T.R.S. Enumeration of eulerian and unicursal planar maps. Discrete Math., 282: 209–221 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  12. Liskovets, V.A., Walsh, T.R.S. Enumeration of unrooted maps on the plane, Rapport technique. UQAM, No. 2005-01, Montreal, Canada, 2005

    Google Scholar 

  13. Liu, Y.P. On functional equation arising from map enumerations. Discrete Math., 123: 93–109 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  14. Liu, Y.P. On the number of Eulerian planar maps. Acta Math. Sci., 12: 418–423 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  15. Liu, Y.P. Enumeration of rooted bipartite planar maps. Acta Math. Sci., 9: 21–28 (1989) (in Chinese)

    Article  Google Scholar 

  16. Liu, Y.P. Enumerative Theory of Maps. Kluwer, Boston, 1999

    MATH  Google Scholar 

  17. Liu, Y.P. A polyhedral theory on graphs. Acta Math. Sin., 10: 136–142 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  18. Liu, Y.P. Combinatorial invariants on graphs. Acta Math. Sin., 11: 211–220 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  19. Liu, Y.P. On the number of rooted c-nets. J. Combin. Theory Ser. B, 36: 118–123 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  20. Long, S.D., Cai, J.L. Enumeration of rooted unicursal planar maps (in Chinese). Acta Math. Sin., 53: 9–16 (2010)

    MATH  Google Scholar 

  21. Mullin, R.C., Schellenberg, P.J. The enumeration of c-nets via quadrangulations. J. Combin. Theory, 4: 256–276 (1964)

    MathSciNet  MATH  Google Scholar 

  22. Ren, H., Liu, Y.P. The number of loopless 4-regular maps on the projective plane. J. Combin. Theory Ser. B, 84: 84–99 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  23. Ren, H., Liu, Y.P., Li, Z.X. Enumeration of 2-connected loopless 4-regular maps on the plane. European J. Combin., 23: 93–111 (2002)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  26. Walsh, T.R.S. Hypermaps versus bipartite maps. J. Combin. Theory Ser. B, 18: 155–163 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  27. Whittaker, E.T., Watson, G.N. A course of modern analysis. Cambridge, 1940

    MATH  Google Scholar 

Download references

Acknowledgements

The authors are grateful to the referees for their careful reading of this paper. Their constructive suggestions enabled us to make some major revisions which made the paper more readable and concrete.

Author information

Authors and Affiliations

  1. Department of Mathematics, Chongqing University of Arts and Sciences, Chongqing, 402160, China

    Shu-de Long

  2. School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, China

    Jun-liang Cai

Authors
  1. Shu-de Long
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jun-liang Cai
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Shu-de Long.

Additional information

Supported by the National Natural Science Foundation of China (No. 10271017,11371133,11571044), the Natural Science Foundation Project of Chongqing (No. cstc2012jjA00041,cstc2014jcyjA00041) and the Innovation Foundation of Chongqing (No. KJTD201321).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Long, Sd., Cai, Jl. Counting rooted 4-regular unicursal planar maps. Acta Math. Appl. Sin. Engl. Ser. 33, 909–918 (2017). https://doi.org/10.1007/s10255-017-0706-x

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10255-017-0706-x

Keywords

2000 MR Subject Classification

Search

Navigation