Enumeration of rooted 3-connected bipartite planar maps
Fuente:
arXiv
Salvato in:
| Autori principali: | , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2022
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866913219259400192 |
|---|---|
| author | Noy, Marc Requilé, Clément Rué, Juanjo |
| author_facet | Noy, Marc Requilé, Clément Rué, Juanjo |
| contents | 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 $t \cdot n^{-5/2} γ^n$, where $γ=ρ^{-1} \approx 2.40958$ and $ρ\approx 0.41501$ is an algebraic number of degree 10. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2202_13929 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Enumeration of rooted 3-connected bipartite planar maps Noy, Marc Requilé, Clément Rué, Juanjo Combinatorics 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 $t \cdot n^{-5/2} γ^n$, where $γ=ρ^{-1} \approx 2.40958$ and $ρ\approx 0.41501$ is an algebraic number of degree 10. |
| title | Enumeration of rooted 3-connected bipartite planar maps |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2202.13929 |