论文标题
枚举生根的三连接的两部分平面图
Enumeration of rooted 3-connected bipartite planar maps
论文作者
论文摘要
我们为计算根生3个连接的两部分平面图的问题提供了第一个解决方案。我们的起点是根据边缘和单色边的数量列举了双色平面地图，在Bernardi和Bousquet-Mélou之后[J.梳子。理论ser。 B，101（2011），315-377]。地图分解为2和3相互连接的组件，使我们能够获得2和3连接的双色图的生成函数。设置为零，将可变标记单色边缘设置为零，我们获得了3个连接的两部分图的生成函数，该图是26的代数。我们从中推断出一个渐近估计值，该估计值是3型连接的两部分平面图的数量2.40958 $和$ρ\大约0.41501 $是学位10的代数数。
We provide the first solution to the problem of counting rooted 3-connected bipartite planar maps. Our starting point is the enumeration of bicoloured planar maps according to the number of edges and monochromatic edges, following Bernardi and Bousquet-Mélou [J. Comb. Theory Ser. B, 101 (2011), 315-377]. The decomposition of a map into 2- and 3-connected components allows us to obtain the generating functions of 2-and 3-connected bicoloured maps. Setting to zero the variable marking monochromatic edges we obtain the generating function of 3-connected bipartite maps, which is algebraic of degree 26. We deduce from it an asymptotic estimate for the number of 3-connected bipartite planar maps of the form $t \cdot n^{-5/2} γ^n$, where $γ=ρ^{-1} \approx 2.40958$ and $ρ\approx 0.41501$ is an algebraic number of degree 10.
