A pattern for the asymptotic number of rooted maps on surfaces
Let \(T_ g(n)\) \((P_ g(n))\) be the number of \(n\)-edged rooted maps (in a certain class) on an orientable (non-orientable) surface of type \(g\), and let \(t_ g\) and \(p_ g\) be the positive constants defined in [\textit{E. A. Bender} and \textit{E. R. Canfield}, J. Comb. Theory, Ser. A 43, 244- 257 (1986; Zbl 0606.05031)]. In [J. Comb. Theory, Ser. A 49, No. 2, 370- 380 (1988; Zbl 0657.05037)], E. A. Bender and N. C. Wormald observed the following pattern: \[ T_ g(n)\sim t_ g (\beta n)^{5(g-1)/2} \gamma^ n,\quad P_ g(n) \sim p_ g (\beta n)^{5(g-1)/2} \gamma^ n, \] for all maps, 2-connected maps and smooth maps. In this paper, we show that many classes of maps fit the following modified pattern: \[ T_ g(n) \sim \alpha t_ g (\beta n)^{5(g-1)/2} \gamma^ n,\quad P_ g(n) \sim \alpha p_ g (\beta_ n)^{5(g-1)/2} \gamma^ n. \]
scientific article; zbMATH DE number 540359
scientific article