Enumeration of rooted 3-connected bipartite planar maps

TL;DR

This paper presents the first enumeration of rooted 3-connected bipartite planar maps using decomposition techniques, resulting in algebraic generating functions and asymptotic estimates for their counts.

Contribution

It introduces a novel enumeration method for 3-connected bipartite planar maps based on decomposition into components, providing algebraic generating functions and asymptotic formulas.

Findings

Generated algebraic degree 26 for the maps' generating function

Derived asymptotic estimate with exponential growth rate approximately 2.40958

Established the first enumeration formula for these maps

Abstract

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-M\'elou [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 $t \cdot n^{-5/2} \gamma^n$, where $\gamma=\rho^{-1} \approx 2.40958$ and $\rho \approx 0.41501$ is an algebraic number of degree 10.