Enumeration of unrooted maps of a given genus

• Authors: Alexander Mednykh;Roman Nedela
• Venue: Journal of Combinatorial Theory Series B
• Year: 2006

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 Θγ (e) of unrooted maps on an orientable surface Sγ of a given genus γ and with a given number of edges e. It has a form of a linear combination Σi,jci,jNgj (fi) of numbers of rooted maps Ngj (fi) for some gj ≤ γ and fi ≤ e. The coefficients ci,j are functions of γ and e. We consider the quotient Sγ/Zl of Sγ by a cyclic group of automorphisms Zl as a two-dimensional orbifold O. The task of determining ci,j requires solving the following two subproblems: (a) to compute the number Epio (Γ, Zl) of order-preserving epimorphisms from the fundamental group Γ of the orbifold O=Sγ/Zl onto Zl; (b) to calculate the number of rooted maps on the orbifold O which lifts along the branched covering Sγ → Sγ/Zl to maps on Sγ with the given number e of edges.The number Epio (Γ, Zl) is expressed in terms of classical number-theoretical functions. The other problem is reduced to the standard enumeration problem of determining the numbers Ng(f) for some g ≤ γ and f ≤ e. It follows that Θγ (e) can be calculated whenever the numbers Ng (f) are known for g ≤ γ and f ≤ e. In the end of the paper the above approach is applied to derive the functions Θγ(e) explicitly for γ ≤ 3. We note that the function Θγ(e) was known only for γ = 0 (Liskovets, 1981). Tables containing the numbers of isomorphism classes of maps with up to 30 edges for genus γ = 1, 2, 3 are presented.