Enumeration of rooted 3-connected bipartite planar maps
We provide the first solution to the problem of counting rooted 3-connected bipartite planar maps. Our starting point is the enumeration of bicoloured planar maps according to the number of edges and monochromatic edges, following Bernardi and Bousquet-Mélou [J. Comb. Theory Ser. B, 101 (2011), 315–...
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Académie des sciences
2024-03-01
|
| Series: | Comptes Rendus. Mathématique |
| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.548/ |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1825206241321811968 |
|---|---|
| author | Noy, Marc Requilé, Clément Rué, Juanjo |
| author_facet | Noy, Marc Requilé, Clément Rué, Juanjo |
| author_sort | Noy, Marc |
| collection | DOAJ |
| description | We provide the first solution to the problem of counting rooted 3-connected bipartite planar maps. Our starting point is the enumeration of bicoloured planar maps according to the number of edges and monochromatic edges, following Bernardi and Bousquet-Mélou [J. Comb. Theory Ser. B, 101 (2011), 315–377]. The decomposition of a map into 2- and 3-connected components allows us to obtain the generating functions of 2- and 3-connected bicoloured maps. Setting to zero the variable marking monochromatic edges we obtain the generating function of 3-connected bipartite maps, which is algebraic of degree 26. We deduce from it an asymptotic estimate for the number of 3-connected bipartite planar maps of the form $t\, n^{-5/2}\gamma ^n$, where $\gamma = \rho ^{-1} \approx 2.40958$ and $\rho \approx 0.41501$ is an algebraic number of degree $10$. |
| format | Article |
| id | doaj-art-7fb1d5ed6d6f4222961c16b57ae7a89a |
| institution | Kabale University |
| issn | 1778-3569 |
| language | English |
| publishDate | 2024-03-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mathématique |
| spelling | doaj-art-7fb1d5ed6d6f4222961c16b57ae7a89a2025-02-07T11:16:25ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692024-03-01362G214315810.5802/crmath.54810.5802/crmath.548Enumeration of rooted 3-connected bipartite planar mapsNoy, Marc0Requilé, Clément1Rué, Juanjo2Centre de Recerca Matemàtica (CRM), Barcelona, Spain; Departament de Matemàtiques and Institut de Matemàtiques (IMTech) de la Universitat Politècnica de Catalunya (UPC), Barcelona, SpainDepartament de Matemàtiques and Institut de Matemàtiques (IMTech) de la Universitat Politècnica de Catalunya (UPC), Barcelona, SpainCentre de Recerca Matemàtica (CRM), Barcelona, Spain; Departament de Matemàtiques and Institut de Matemàtiques (IMTech) de la Universitat Politècnica de Catalunya (UPC), Barcelona, SpainWe provide the first solution to the problem of counting rooted 3-connected bipartite planar maps. Our starting point is the enumeration of bicoloured planar maps according to the number of edges and monochromatic edges, following Bernardi and Bousquet-Mélou [J. Comb. Theory Ser. B, 101 (2011), 315–377]. The decomposition of a map into 2- and 3-connected components allows us to obtain the generating functions of 2- and 3-connected bicoloured maps. Setting to zero the variable marking monochromatic edges we obtain the generating function of 3-connected bipartite maps, which is algebraic of degree 26. We deduce from it an asymptotic estimate for the number of 3-connected bipartite planar maps of the form $t\, n^{-5/2}\gamma ^n$, where $\gamma = \rho ^{-1} \approx 2.40958$ and $\rho \approx 0.41501$ is an algebraic number of degree $10$.https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.548/ |
| spellingShingle | Noy, Marc Requilé, Clément Rué, Juanjo Enumeration of rooted 3-connected bipartite planar maps Comptes Rendus. Mathématique |
| title | Enumeration of rooted 3-connected bipartite planar maps |
| title_full | Enumeration of rooted 3-connected bipartite planar maps |
| title_fullStr | Enumeration of rooted 3-connected bipartite planar maps |
| title_full_unstemmed | Enumeration of rooted 3-connected bipartite planar maps |
| title_short | Enumeration of rooted 3-connected bipartite planar maps |
| title_sort | enumeration of rooted 3 connected bipartite planar maps |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.548/ |
| work_keys_str_mv | AT noymarc enumerationofrooted3connectedbipartiteplanarmaps AT requileclement enumerationofrooted3connectedbipartiteplanarmaps AT ruejuanjo enumerationofrooted3connectedbipartiteplanarmaps |