Almost All Rooted Maps Have Large Representativity
نویسندگان
چکیده
Let M be a map on a surface S. The edge-width of M is the length of a shortest non-contractible 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 de ned to be in nity.) A map is an 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 logn and 2. the fraction of such maps which are LEW-embeddings and the fraction which are not LEW-embeddings both exceed n c2. 2
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