论文标题
标记为4型平面图的枚举II：渐近学
Enumeration of labelled 4-regular planar graphs II: asymptotics
论文作者
论文摘要
这项工作是对文章的后续行动[Proc。\伦敦数学。在本文中，我们获得了$ n $顶点上标记为4型平面图的$ g_n $的精确渐近估计。我们的估计是$ g_n \ sim g \ cdot n^{ - 7/2}ρ{ - n} n！$的形式，其中$ g> 0 $是常数，$ρ\ of 0.24377 $是生成函数$ \ sum_ ge 0} g_ ge n} g_n x x^n/n！列举几类平面图。除了分析方法外，我们的解决方案还需要大量使用计算机代数，以便处理大型多元多项式方程式。我们还获得了2和3连接的4个规则平面图的渐近估计值，以及连接和2连接的4型简单映射的数量。
This work is a follow-up of the article [Proc.\ London Math.\ Soc.\ 119(2):358--378, 2019], where the authors solved the problem of counting labelled 4-regular planar graphs. In this paper, we obtain a precise asymptotic estimate for the number $g_n$ of labelled 4-regular planar graphs on $n$ vertices. Our estimate is of the form $g_n \sim g\cdot n^{-7/2} ρ^{-n} n!$, where $g>0$ is a constant and $ρ\approx 0.24377$ is the radius of convergence of the generating function $\sum_{n\ge 0}g_n x^n/n!$, and conforms to the universal pattern obtained previously in the enumeration of several classes of planar graphs. In addition to analytic methods, our solution needs intensive use of computer algebra in order to deal with large systems of multivariate polynomial equations. We also obtain asymptotic estimates for the number of 2- and 3-connected 4-regular planar graphs, and for the number of 4-regular simple maps, both connected and 2-connected.
