ALMOST-ALL ROOTED MAPS HAVE LARGE REPRESENTATIVITY

Let M be a map on a surface S. The edge-width of M is the length of a shortest noncontractible cycle of M. The face-width (or, representativity) of M is the smallest number of intersections a noncontractible curve in S has with M. (The edge-width and face-width of a planar map may be defined to be infinity.) A map is a large-edge-width embedding (LEW-embedding) if its maximum face valency is less than its edge-width. For several families of rooted maps on a given surface, we prove that there are positive constants c1 and c2, depending on the family and the surface, such that 1. almost all maps with n edges have face-width and edge-width greater than c1 log n, and 2. the fraction of such maps that are LEW-embeddings and the fraction that are not LEW-embeddings both exceed n-c2 (C) 1994 John Wiley & Sons, Inc.