Enumeration of simple bipartite maps on the sphere and the projective plane

AbstractThis paper is concerned with the number of rooted simple bipartite maps on the plane according to the root-face valencies and the number of edges of the maps. An explicit formula with one parameter is given. Furthermore, a special kind of rooted simple bipartite maps on the projective plane are counted and, as special cases, recursive formulae for maps with fewer faces on the projective plane are presented as well

Submaps of maps. I. General 0–1 laws

AbstractLet Mn be the set of n edge maps of some class on a surface of genus g. When g = 0 (planar maps) we show how to prove that limn → ∞ |Mn|1/n exists for many classes of maps. Let P be a particular map that can appear as a submap of maps in our class. There is often a strong 0–1 law for the property that P is a submap of a randomly chosen map in Mn: If P is planar, then almost all Mn contain at least cn disjoint copies of P for small enough c; while if P is not planar, almost no Mn contain a copy of P. We show how to establish this for various classes of maps. For planar P, the existence of limn → ∞ |Mn|1n suffices. For nonplanar P, we require more detailed asymptotic information

A formula for the bivariate map asymptotics constants in terms of the univariate map asymptotics constants

The parameters tg, pg, tg(r) and pg(r) appear in the asymptotics for a variety of maps on surfaces and embeddable graphs. In this paper we express tg(r) in terms of tg and pg(r) in terms of pg