Asymptotic enumeration of reversible maps regardless of genus

Michael Drmota, Roman Nedela

Abstract


We derive asymptotic expansions for the numbers U(n) of isomorphism classes of sensed maps on orientable surfaces with given number of edges n, where we do not specify the genus and for the numbers A(n) of reflexible maps with n edges. As expected the ratio A(n)/U(n) → 0 for n → ∞. This shows that almost all maps are chiral. Moreover, we show log A(n) ∼ (1/2)log U(n) ∼ (n/2)log n. Due to a correspondence between sensed maps with given number of edges and torsion-free subgroups of the group Γ = < x, y | y2 = 1 > of given index, the obtained results give an information on asymptotic expansions for the number of conjugacy classes of such subgroups of given index.

Keywords


Graph, Map, Enumeration, Asymptotic.

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DOI: https://doi.org/10.26493/1855-3974.145.42c

ISSN: 1855-3974

Issues from Vol 6, No 1 onward are partially supported by the Slovenian Research Agency from the Call for co-financing of scientific periodical publications