The asymptotic numbers of certain kinds of regular toroidal maps

The asymptotic number (of isomorphism classes) of toroidal maps of type (3; 6) with at most n vertices is found, together with the fraction of those with multiplicity 1. Accurate lower and upper asymptotic estimates are provided for the number of toroidal maps of type (3; 6) with a Hamiltonian normal cycle and with at most n vertices. The case of type (6; 3) toroidal maps follows by duality. Similar results are obtained for toroidal maps of type (4; 4). (Type (p; q)= partition into p-gons, q edges incident to each vertex; normal cycle in a map of type (3; 6) = a cycle that leaves, at each of its vertices, exactly two edges on the right; multiplicity of a toroidal map of type (3; 6) = the greatest common divisor of the numbers of the three kinds of normal cycles.) c © 2000 Elsevier Science B.V. All rights reserved.