Enumeration of labelled 4-regular planar graphs II: Asymptotics. (May 2023)

• Record Type: Journal Article
• Authors: Noy, Marc; Requilé, Clément; Rué, Juanjo
• Is Part Of: European journal of combinatorics. Volume 110(2023)
• Publication Date: 2023-05
• Subjects: Combinatorial analysis -- Periodicals; Analyse combinatoire -- Périodiques; Combinatorial analysis; Periodicals; Electronic journals; 511.6
• DOI: 10.1016/j.ejc.2022.103661
• Languages: English
• ISSNs: 0195-6698

Abstract

Abstract: This work is a follow-up of the article (Noy et al., 2019), where the authors solved the problem of counting labelled 4-regular planar graphs. In this paper, we obtain a precise asymptotic estimate for the number g n of labelled 4-regular planar graphs on n vertices. Our estimate is of the form g n ∼ g ⋅ n − 7 / 2 ρ − n n !, where g > 0 is a constant and ρ ≈ 0 . 24377 is the radius of convergence of the generating function ∑ n ≥ 0 g n x n / n !, and conforms to the universal pattern obtained previously in the enumeration of several classes of planar graphs. In addition to analytic methods, our solution needs intensive use of computer algebra in order to deal with large systems of multivariate polynomial equations. We also obtain asymptotic estimates for the number of 2- and 3-connected 4-regular planar graphs, and for the number of 4-regular simple maps, both connected and 2-connected.