The Genus Distribution of Cubic Graphs and Asymptotic Number of Rooted Cubic Maps with High Genus

Let Cn,g be the number of rooted cubic maps with $2n$ vertices on the orientable surface of genus g. We show that the sequence (Cn,g:g≥ 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 Cn,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 3/π\,n!6ⁿ.