Abstract
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.
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