Enumeration of rooted 3-connected bipartite planar 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-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 tn -5/2 γ n , where γ=ρ -1 ≈2.40958 and ρ≈0.41501 is an algebraic number of degree 10.