Enumeration of unrooted maps of a given genus

Let Ng(f ) denote the number of rooted maps of genus g having f edges. An exact formula for Ng(f ) is known for g = 0 (Tutte, 1963), g = 1 (Arques, 1987), g = 2,3 (Bender and Canfield, 1991). In the present paper we derive an enumeration formula for the number (e) of unrooted maps on an orientable surface S of a given genus and with a given number of edges e. It has a form of a linear combination i,j ci,j Ngj (fi) of numbers of rooted maps Ngj (fi) for some gj and fi e. The coefficients ci,j are functions of and e. We consider the quotient S /Z of S by a cyclic group of automorphisms Z as a two-dimensional orbifold O. The task of determining ci,j requires solving the following two subproblems: (a) to compute the number Epio(,Z ) of order-preserving epimorphisms from the fundamental group of the orbifold O = S /Z onto Z ; (b) to calculate the number of rooted maps on the orbifold O which lifts along the branched covering S S /Z to maps on S with the given number e of edges. The nu...

• Type: Journal
• Year: 2006
• Where: JCT
• Authors: Alexander Mednykh, Roman Nedela
• Keywords: Enumeration Formula, Exact Formula, JCT 2006, Unrooted Maps