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

TL;DR

This paper investigates the distribution of cubic maps on surfaces of various genera, establishing asymptotic normality and deriving formulas for their counts, including the total number of rooted cubic maps with 2n vertices.

Contribution

It provides the first asymptotic normality results for the genus distribution of rooted cubic maps and derives explicit formulas for their enumeration.

Findings

The genus distribution of rooted cubic maps is asymptotically normal.

Mean and variance of the genus distribution grow logarithmically with n.

Total number of rooted cubic maps with 2n vertices is asymptotically .477 n! 6^n.

Abstract

Let $C_{n,g}$ be the number of rooted cubic maps with $2n$ vertices on the orientable 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 $(0,2)$. Using rotation systems and Bender's theorem about generating functions with fast-growing coefficients, we derive simple asymptotic expressions for the numbers of rooted regular maps, disregarding the genus. In particular, we show that the number of rooted cubic maps with $2n$ vertices, disregarding the genus, is asymptotic to $\frac{3}{\pi}\,n!6^n$.