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Enumeration of rooted 3-connected bipartite planar maps

Enumeration of rooted 3-connected bipartite planar maps AbstractWe 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Pure Mathematics and Applications de Gruyter

Enumeration of rooted 3-connected bipartite planar maps

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Publisher
de Gruyter
Copyright
© 2022 Marc Noy et al., published by Sciendo
eISSN
1788-800X
DOI
10.2478/puma-2022-0015
Publisher site
See Article on Publisher Site

Abstract

AbstractWe 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.

Journal

Pure Mathematics and Applicationsde Gruyter

Published: Jun 1, 2022

Keywords: enumeration; generating function; asymptotics; bipartite planar map; 05A15; 05A16

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