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Enumeration of unrooted orientable maps of arbitrary genus by number of edges and vertices. (English) Zbl 1246.05076
Summary: A genus-\(g\) map is a 2-cell embedding of a connected graph on a closed, orientable surface of genus \(g\) without boundary, that is, a sphere with \(g\) 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-\(g\) maps, we first needed to count rooted maps of every genus up to \(g\). A recursively-defined generating function for counting rooted maps of arbitrary genus by number of faces and vertices was found by D. Arquès and A. Giorgetti [J. Comb. Theory, Ser. B 77, No. 1, 1–24 (1999; Zbl 1029.05073)], and numerical values were obtained for \(g\leq 3\). The first two authors jointly extended the solution of this recursion up to g=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. A. Mednykh and R. Nedela [J. Comb. Theory, Ser. B 96, No. 5, 706–729 (2006; Zbl 1102.05033)] 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 \(g\)=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.

05C30 Enumeration in graph theory
05C85 Graph algorithms (graph-theoretic aspects)
Full Text: DOI
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