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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 (2011)]. 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 <italic>t</italic> · <italic>n</italic>−5/2<italic>γ</italic><italic>n</italic>, where <italic>γ</italic> = <italic>ρ</italic>−1 ≈ 2.40958 and <italic>ρ</italic> ≈ 0.41501 is an algebraic number of degree 10.
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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 (2011)]. 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−5/2γn, where γ = ρ−1 ≈ 2.40958 and ρ ≈ 0.41501 is an algebraic number of degree 10.

Enumeration of rooted 3-connected bipartite planar maps

Departament de Matemàtiques and Institut de Matemàtiques

Universitat Politècnica de Catalunya, and Centre de Recerca Matemàtica (CRM)

Pure Mathematics and Applications

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

{"article-title":"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...

Enumeration of rooted 3-connected bipartite planar maps

Published Online: Jun 18, 2022

Page range: 97 - 105

Received: Mar 31, 2022

Accepted: May 15, 2022

DOI: https://doi.org/10.2478/puma-2022-0015

Keywords
enumeration, generating function, asymptotics, bipartite planar map

© 2022 Marc Noy et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Enumeration of rooted 3-connected bipartite planar maps

Published Online: Jun 18, 2022

Page range: 97 - 105

Received: Mar 31, 2022

Accepted: May 15, 2022

DOI: https://doi.org/10.2478/puma-2022-0015

Keywordsenumeration, generating function, asymptotics, bipartite planar map

© 2022 Marc Noy et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Keywords
enumeration, generating function, asymptotics, bipartite planar map