Map enumeration

Roman Nedela (Matej Bel University, Slovakia)

History of map enumeration goes back to Tutte's results in years 1961-1963 when he derived a formula for the number of rooted planar maps with a given number of edges as well as for several other distinguished classes of maps. Since then there were more than 100 contribution to the theory of map enumeration, however, most of them counting rooted maps. Except the results of Liskovets and Wormald - they derived the formula for the number of isoclasses of spherical maps, there were no results on the number of isomorphism classes of maps. In 2006 we have published a paper in J. Comb. Theory B, where a new approach to the enumeration problem is proposed. As a result we derived formulae for the number of isoclasses of maps of genera up to 4. The new geometric approach turned to be useful also in solving other problems in topological graph theory, coverings of manifolds and combinatorics. In our talk we first explain the new approach and secondly, show how it can be used to solve several different problems in combinatorics, number theory, topology...

Joint work with A. Mednykh.