论文标题
立方图的属分布和具有高属的根分立图的渐近数
The genus distribution of cubic graphs and asymptotic number of rooted cubic maps with high genus
论文作者
论文摘要
令$ c_ {n，g} $为$ 2N $顶点的根cutib table属于$ g $的$ 2N $顶点的数量。我们表明序列$（c_ {n，g}：g \ ge 0）$在平均值和方差渐近均为$（1/2）（n- \ ln n）$和$（1/4）\ ln n $。当$（n-2g）/\ ln n $均在$（0,2）$的任何封闭的子间隙中时，我们将$ c_ {n，g} $得出渐近表达式。使用旋转系统和Bender的定理来生成具有快速增长系数的函数，我们为根的常规地图数量得出了简单的渐近表达式，而无视该属。特别是，我们表明，无视该属的$ 2N $顶点的根生物图的数量是渐近的，对$ \ frac {3}π\，n！6^n $。
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 $\frac{3}π\,n!6^n$.
