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Let $C_{n,g}$ be the number of rooted cubic maps with $2n$ vertices on theorientable surface of genus $g$. We show that the sequence $(C_{n,g}:g\ge 0)$is asymptotically normal with mean and variance asymptotic to $(1/2)(n-\ln n)$and $(1/4)\ln n$, respectively. We derive an asymptotic expression for$C_{n,g}$ when $(n-2g)/\ln n$ lies in any closed subinterval of $(0,2)$. Usingrotation systems and Bender's theorem about generating functions withfast-growing coefficients, we derive simple asymptotic expressions for thenumbers of rooted regular maps, disregarding the genus. In particular, we showthat the number of rooted cubic maps with $2n$ vertices, disregarding thegenus, is asymptotic to $\frac{3}{\pi}\,n!6^n$.

The genus distribution of cubic graphs and asymptotic number of rooted cubic maps with high genus