Geodesic
On the enumeration of hypermaps which are self-equivalent with respect to reversing the colors of vertices
Zapiski Nauchnykh Seminarov POMI, Combinatorics and graph theory. Part V, Tome 446 (2016), pp. 31-39 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

A map $(S,G)$ is a closed Riemann surface $S$ with an embedded graph $G$ such that $S\setminus G$ is the disjoint union of connected components, called faces, each of which is homeomorphic to an open disk. Tutte began a systematic study of maps in the 1960s, and contemporary authors are actively developing it. We recall the concept of a circular map introduced by the author and Mednykh and demonstrate a relationship between bipartite maps and circular maps through the concept of the duality of maps. We thus obtain an enumeration formula for the number of bipartite maps with a given number of edges. A hypermap is a map whose vertices are colored black and white in such a way that every edge connects vertices of different colors. Hypermaps are also known as dessins d'enfants (or Grothendieck's dessins). A hypermap is self-equivalent with respect to reversing the colors of vertices if it is equivalent to the hypermap obtained by reversing the colors of its vertices. The main result of this paper is an enumeration formula for the number of unrooted hypermaps, regardless of genus, which have $n$ edges and are self-equivalent with respect to reversing the colors of vertices. 
@article{ZNSL_2016_446_a2,
     author = {M. Deryagina},
     title = {On the enumeration of hypermaps which are self-equivalent with respect to reversing the colors of vertices},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {31--39},
     year = {2016},
     volume = {446},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2016_446_a2/}
}
TY  - JOUR
AU  - M. Deryagina
TI  - On the enumeration of hypermaps which are self-equivalent with respect to reversing the colors of vertices
JO  - Zapiski Nauchnykh Seminarov POMI
PY  - 2016
SP  - 31
EP  - 39
VL  - 446
UR  - http://geodesic.mathdoc.fr/item/ZNSL_2016_446_a2/
LA  - en
ID  - ZNSL_2016_446_a2
ER  - 
%0 Journal Article
%A M. Deryagina
%T On the enumeration of hypermaps which are self-equivalent with respect to reversing the colors of vertices
%J Zapiski Nauchnykh Seminarov POMI
%D 2016
%P 31-39
%V 446
%U http://geodesic.mathdoc.fr/item/ZNSL_2016_446_a2/
%G en
%F ZNSL_2016_446_a2
M. Deryagina. On the enumeration of hypermaps which are self-equivalent with respect to reversing the colors of vertices. Zapiski Nauchnykh Seminarov POMI, Combinatorics and graph theory. Part V, Tome 446 (2016), pp. 31-39. http://geodesic.mathdoc.fr/item/ZNSL_2016_446_a2/

[1] R. Cori, Un code pour les graphes planaires et ses applications, Thèse de Doctorat, Paris, 1973

[2] A. Grothendieck, “Esquisse d'un programme (1984)”, Geometric Galois Action, v. 1, Around Grothendieck's Esquisse d'un Programme, eds. Schneps L., Lochak P., Cambridge Univ. Press, 1997, 243–283 | DOI | MR | Zbl

[3] M. Hall (Jr.), “Subgroups of finite index in free groups”, Canadian J. Math., 1 (1949), 187–190 | DOI | MR | Zbl

[4] D. M. Jackson, T. I. Visentin, An Atlas of the Smaller Maps in Orientable and Nonorientable Surfaces, Chapman Hall/CRC Press, Boca Raton, 2001 | MR | Zbl

[5] S. K. Lando, A. K. Zvonkin, Graphs on Surfaces and Their Applications, With Appendix by Don B. Zagier, Springer-Verlag, 2004 ; A. K. Zvonkin, S. K. Lando, Grafy na poverkhnostyakh i ikh prilozheniya, Izd-vo MTsNMO, M., 2010 | MR | Zbl

[6] V. A. Liskovets, “Enumerative formulae for unrooted planar maps: A pattern”, Electronic J. Combin., 11:1 (2004), Research paper R88, 14 pp. | MR | Zbl

[7] V. A. Liskovets, T. R. S Walsh, “Enumeration of Eulerian and unicursal planar maps”, Discr. Math., 282 (2004), 209–221 | DOI | MR | Zbl

[8] A. Mednykh, R. Nedela, “Enumeration of unrooted hypermaps of a given genus”, Discr. Math., 310:3 (2010), 518–526 | DOI | MR | Zbl

[9] W. T. Tutte, “A census of planar maps”, Canadian J. Math., 15 (1963), 249–271 | DOI | MR | Zbl

[10] T. R. S. Walsh, “Hypermaps versus bipartite maps”, J. Combin. Theory, Ser. B, 18:2 (1975), 155–163 | DOI | MR | Zbl

[11] On-Line Encyclopedia of Integer Sequences, http://www.oeis.org

[12] M. A. Deryagina, A. D. Mednykh, “On the enumeration of circular maps with given number of edges”, Sib. Math. J., 54:4 (2013), 624–639 | DOI | MR | Zbl

[13] V. A. Liskovec, “On the enumeration of subgroups of a free group”, Dokl. Akad. Nauk BSSR, 15:1 (1971), 6–9