Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-25T18:14:26.685Z Has data issue: false hasContentIssue false
  • English
  • Français

Asymptotic Enumeration of Constellations and Related Families of Maps on Orientable Surfaces

Published online by Cambridge University Press:  01 July 2009

GUILLAUME CHAPUY
GUILLAUME CHAPUY*
Affiliation:
Laboratoire d'Informatique de l'École Polytechnique, 91128 Palaiseau Cedex, France (e-mail: [email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

We perform the asymptotic enumeration of two classes of rooted maps on orientable surfaces: m-hypermaps and m-constellations. For m = 2 they correspond respectively to maps with even face degrees and bipartite maps. We obtain explicit asymptotic formulas for the number of such maps with any finite set of allowed face degrees.

Our proofs combine a bijective approach, generating series techniques related to lattice walks, and elementary algebraic graph theory.

A special case of our results implies former conjectures of Z. Gao.

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 18 , Issue 4 , July 2009 , pp. 477 - 516
Copyright
Copyright © Cambridge University Press 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Banderier, C. and Flajolet, P. (2002) Basic analytic combinatorics of directed lattice paths. Theoret. Comput. Sci. 281 37–80.CrossRefGoogle Scholar
[2]Banderier, C., Flajolet, P., Schaeffer, G. and Soria, M. (2001) Random maps, coalescing saddles, singularity analysis, and Airy phenomena. Random Struct. Alg. 19 194–246.CrossRefGoogle Scholar
[3]Bender, E. (1991) Some unsolved problems in map enumeration. Bull. Inst. Combin. Appl. 3 51–56.Google Scholar
[4]Bender, E. A. and Canfield, E. R. (1986) The asymptotic number of rooted maps on a surface. J. Combin. Theory Ser. A 43 244–257.CrossRefGoogle Scholar
[5]Bender, E. A. and Canfield, E. R. (1991) The number of rooted maps on an orientable surface. J. Combin. Theory Ser. B 53 293–299.CrossRefGoogle Scholar
[6]Bousquet-Mélou, M. (2008) Discrete excursions. Sém. Lothar. Combin. 57.Google Scholar
[7]Bousquet-Mélou, M. and Schaeffer, G. (2000) Enumeration of planar constellations. Adv. Appl. Math. 24 337–368.CrossRefGoogle Scholar
[8]Bouttier, J., Di Francesco, P. and Guitter, E. (2004) Planar maps as labeled mobiles. Electron. J. Combin. 11 # 69 (electronic).CrossRefGoogle Scholar
[9]Bouttier, J. and Guitter, E. (2008) Statistics of geodesics in large quadrangulations. J. Phys. A 41 no. 14.CrossRefGoogle Scholar
[10]Chapuy, G., Marcus, M. and Schaeffer, G. (2007) A bijection for rooted maps on orientable surfaces. arXiv:0712.3649v1 [math.CO].Google Scholar
[11]Chassaing, P. and Schaeffer, G. (2004) Random planar lattices and integrated superBrownian excursion. Probab. Theory Rel. Fields 128 161–212.CrossRefGoogle Scholar
[12]Cori, R. and Vauquelin, B. (1981) Planar maps are well labeled trees. Canad. J. Math. 33 1023–1042.CrossRefGoogle Scholar
[13]Drmota, M. (1997) Systems of functional equations. Random Struct. Alg. 10 103–124.3.0.CO;2-Z>CrossRefGoogle Scholar
[14]Flajolet, P. and Odlyzko, A. (1990) Singularity analysis of generating functions. SIAM J. Discrete Math. 3 216–240.CrossRefGoogle Scholar
[15]Gao, Z. (1993) The number of degree restricted maps on general surfaces. Discrete Math. 123 47–63.CrossRefGoogle Scholar
[16]Lando, S. K. and Zvonkin, A. K. (2004) Graphs on Surfaces and their Applications, Vol. 141 of Encyclopaedia of Mathematical Sciences, Springer, Berlin.CrossRefGoogle Scholar
[17]Le Gall, J.-F. (2007) The topological structure of scaling limits of large planar maps. Inventiones Mathematica 169 621–670.CrossRefGoogle Scholar
[18]Le Gall, J.-F. (2008) Geodesics in large planar maps and in the Brownian map. arXiv:0804.3012v1.Google Scholar
[19]Le Gall, J.-F. and Paulin, F. (2008) Scaling limits of bipartite planar maps are homeomorphic to the 2-sphere. Geom. Funct. Anal. 18 no. 3.CrossRefGoogle Scholar
[20]Marcus, M. and Schaeffer, G. (1999) Une bijection simple pour les cartes orientables. Manuscript: see www.lix.polytechnique.fr/~schaeffe/biblio/.Google Scholar
[21]Miermont, G. (2007) Tessellations of random maps of arbitrary genus. arXiv:0712.3688v1 [math.PR].Google Scholar
[22]Mohar, B. and Thomassen, C. (2001) Graphs on Surfaces, Johns Hopkins Studies in the Mathematical Sciences, Johns Hopkins University Press.CrossRefGoogle Scholar
[23]Schaeffer, G. (1999) Conjugaison d'arbres et cartes combinatoires aléatoires. PhD thesis.Google Scholar
[24]Tutte, W. T. (1962) A census of planar triangulations. Canad. J. Math. 14 21–38.CrossRefGoogle Scholar
[25]Tutte, W. T. (1962) A census of Hamiltonian polygons. Canad. J. Math. 14 402–417.CrossRefGoogle Scholar
[26]Tutte, W. T. (1962) A census of slicings. Canad. J. Math. 14 708–722.CrossRefGoogle Scholar
[27]Tutte, W. T. (1963) A census of planar maps. Canad. J. Math. 15 249–271.CrossRefGoogle Scholar
[28]Tutte, W. T. (1984) Graph Theory, Vol. 21 of Encyclopedia of Mathematics and its Applications, Addison-Wesley Reading, MA.Google Scholar