Asymptotic enumeration of reversible maps regardless of genus - Drmota, Michael (Ars mathematica contemporanea, 2012)
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 ?$\log A(n) \sim \frac{1}{2} \log U(n) \sim \frac{n}{2} \log n$?. Due to a correspondence between sensed maps with given number of edges and torsion-free subgroups of the group ?$Γ = \langle x, y | y^2 = 1 \rangle$? 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: asymptotic; enumeration; grafi; graph; graph theory; map; preštevanje; teorija grafov; zemljevid