Submap Density and Asymmetry Results for Two Parameter Map Families

Abstract

Let [Mscr ]n,k(S) be the set of n-edge k-vertex rooted maps in some class on the surface S. Let P be a planar map in the class. We develop a method for showing that almost all maps in [Mscr ]n,k(S) contain many copies of P. One consequence of this is that almost all maps in [Mscr ]n,k(S) have no symmetries. The classes considered include c-connected maps (c [les ] 3) and certain families of degree restricted maps.