Enumeration of unrooted orientable maps of arbitrary genus by number of edges and vertices

A genus  -ggmap   is a 2-cell embedding of a connected graph on a closed, orientable surface of genus gg without boundary, that is, a sphere with gg handles. Two maps are equivalent if they are related by a homeomorphism between their embedding surfaces that takes the vertices, edges and faces of one map into the vertices, edges and faces, respectively, of the other map, and preserves the orientation of the surfaces. A map is rooted if a dart of the map–half an edge–is distinguished as its root. Two rooted maps are equivalent if they are related by a homeomorphism that has the above properties and that also takes the root of one map into the root of the other. By counting maps, rooted or unrooted, we mean counting equivalence classes of those maps.To count unrooted genus-gg maps, we first needed to count rooted maps of every genus up to gg. A recursively-defined generating function for counting rooted maps of arbitrary genus by number of faces and vertices was found by Didier Arquès and the second author, and numerical values were obtained for g≤3g≤3. The first two authors jointly extended the solution of this recursion up to g=5g=5. Valery Liskovets used quotient maps and Tutte’s enumeration formula for rooted genus-0 maps to count unrooted genus-0 maps by number of edges. The third author and Roman Nedela generalized Liskovets’ method to count unrooted maps of genus 1, 2 and 3 by number of edges.In this paper, we describe the above-mentioned previously-obtained results and present the following new ones. The first author wrote an optimized program in C to extend the solution of the Arquès–Giorgetti recurrence, and thus the enumeration of rooted maps by number of edges and vertices, up to g=10g=10. The second and third authors extended the enumeration of rooted and unrooted maps by number of edges up to genus 11 using tables computed and published on a web site by Ján Karabás, a student of Nedela. Using Liskovets’ refinement of the Mednykh–Nedela method and the tables of numbers of rooted maps, the first author counted unrooted maps of genus up to 10 by number of edges and vertices.