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

Zhicheng Gao¹

¹School of Mathematics and Statistics Carleton University

The electronic journal of combinatorics, Tome 31 (2024) no. 2

Résumé

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 $(0,1)$.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 $\frac{3}{\pi}\,n!6^n$.

Détail

DOI : 10.37236/11533

Classification : 05C10, 05C75, 05A15, 60B05
Mots-clés : rotation systems, Bender's theorem