关于4-连通三角剖分的计数

本文讨论4-连通极大平面地图的计数问题.从地图对偶的角度考虑,它等价 于强3-连通3-正则有根平面地图的计数问题.在此,我们获得了具有一个和两个变 量的精确计数公式.本文的结果简化并推广了文[1,2]中的相应结果.     

带根4-正则单行平面地图的计数(英文)

本文研究了带根4-正则单行平面地图的计数问题,并给出了以其非根点数和两个奇点次为三个参数的一些计数公式.     

高亏格曲面上地图的计数

本文给出了可定向曲面(亏格2,3)和不可定向曲面(亏格5)上根瓣丛以边数为参数时相应的计数显式.与此同时,考虑一类与瓣从拓扑等价的地图类: (无环,简单)近2-正则地图,通过一种组合方法,给出了多参数下平面近2一正则地图的计数显式,亦得到了任意亏格曲面上该类地图的具体个数.     

有根不可分离平面偶地图的计数

本文给出了由边数和根面的次作为指标的有根不可分离平面偶地图的计数函数所满足的函效方程,此方程是三次的。而,有根一般平面偶地图的相应计数函数所满足的方程却是二次的。从此后一个方程出发得到了上面提到的二个计数函数的具体形式。同时,发现了这个有根不可分离平面偶地图的计数函数与Fabonacci叙列的关系。     

有根无环平面地图节点剖分计数方程

一个平面地图,如果无有边是环,则称为是无环的.有根的意义与[1]中的相同.在那里对于此类地图的一些计数问题作了研究,但从未触及到节点剖分.这篇文章的主要目的在于研究这类地图的依节点剖分的计数.求出了有根无环平面地图依节点剖分计数的母函数所满足的一个泛函方程.并且,作为这一方程的一种应用,求出了一类在节点的最大次给定情况下的有根无环平面地图依节点剖分计数的一些结果.     

平面上一般地图的色和与双色和

本文研究了平面上一般带根地图的色和与双色和,得到了这类地图的色和与双色和函数方程。从这类地图的色和函数方程,导出了平面上一般无环地图、平面上二部地图和平面上欧拉地图的计数函数方程。还得到了一些计数函数的计数显式。     

带根无环欧拉平面地图的计数(英文)

自20世纪60年代初Tutte的开创性工作以来,许多学者在带根地图的计数方面作了很多工作,但许多类无环地图的计数仍没有被处理.本文主要研究以根点次、非根点数和内面数为三个参数的带根无环欧拉平面地图的计数问题.     

至多有两个无公共边圈的有根平面地图计数

本文研究至多有两个无公共边圈的有根平面地图,提出了这种地图的节点剖分计数函数和以它的根次、边数和一次点数为三个参数的计数函数所满足方程。     

球面和射影平面上不可分地图的色和

给出了球面和射影平面上带根不可分地图的色和方程,从色和方程导出了球面和射影平面上带根一般不可分地图、二部地图的计数函数方程. 利用色和理论,研究不同类地图的计数问题,得到了一种研究计数问题的新方法. 此外,还得到了一些计数显示表达式.     

近三正则3—连通平面地图的计数

本文提供了便于依根点次,边数和根面次计数近三正则3－连通有根平面地图的一个函数方程,继之得到其参数形式解,并由此通过Lagrange反演导出了它的计数显示,本文推广了[3]和[4]的结果。     

Chromatic sum equations for rooted cubic planar maps

This paper provides a functional equation satisfied by rooted nearly cubic planar maps. By a nearly cubic map is meant such a map that all the vertices have valency 3 with the exception of at most the root-vertex. And, as a consequence, the corresponding functional equation for rooted cubic planar maps is found.     

Simplification of formulas for the number of maps on surfaces

Two combinatorial identities obtained by the author are used to simplify formulas for the number of general rooted cubic planar maps, for the number of g-essential maps on surfaces of small genus, and also for rooted Eulerian maps on the projective plane. Besides, an asymptotics for the number of maps with a large number of vertices is obtained.     

A census of boundary cubic rooted planar maps

In this paper, boundary cubic rooted planar maps are investigated and exact enumerative formulae are given. First, an enumerative formula for boundary cubic inner-forest maps with the size (number of edges) as a parameter is derived. For the special case of boundary cubic inner-tree maps, a simple formula with two parameters is presented. Further, according to the duality, a corresponding result for outer-planar maps is obtained. Finally, some results for boundary cubic planar maps and general planar maps are obtained. Furthermore, two known Tutte's formulae are easily deduced in the paper.     

THE NUMBER OF ROOTED NEARLY CUBIC C-NETS

1. IntroductionW.T. Tutte's original papers[1--3) on the enumerative theory of rooted planar maps havebrought forth a series of papers on enumerating triangulations. The enumeration of generalrooted planar maps has then also been investigated and a number of elegant results havebeen obtained, although relatively fewer than that of triangulations. As the dual case oftriangulations, the enumerative theory of cubic maps has also been developed, though thereare a lot of problems waiting for solut…     

Computing Harmonic Maps and Conformal Maps on Point Clouds

We use a narrow-band approach to compute harmonic maps and conformal maps for surfaces embedded in the Euclidean 3-space, using point cloud data only. Given a surface, or a point cloud approximation, we simply use the standard cubic lattice to approximate its $\epsilon$-neighborhood. Then the harmonic map of the surface can be approximated by discrete harmonic maps on lattices. The conformal map, or the surface uniformization, is achieved by minimizing the Dirichlet energy of the harmonic map while deforming the target surface of constant curvature. We propose algorithms and numerical examples for closed surfaces and topological disks. To the best of the authors' knowledge, our approach provides the first meshless method for computing harmonic maps and uniformizations of higher genus surfaces.     

The number of nonseparable maps on the projective plane

In this paper, we study the rooted nonseparable maps on the sphere and the projective plane with the valency of root-face and the number of edges as parameters. Explicit expression of enumerating functions are obtained for such maps on the sphere and the projective plane. A parametric expression of the generating function is obtained for such maps on the projective plane, from which asymptotic evaluations are derived. Moreover, if the number of edges is sufficiently large, then almost all nonseparable maps on the projective plane are not triangulation.     

The number of rooted circuit boundary maps

In this article, the authors discuss two kinds of new planar maps： pan-fan maps and circuit boundary maps, and provide explicit expressions about their enumerating functions with different parameters. Meanwhile, two explicit counting formulas for circuit cubic boundary maps with two parameters; the size and the valency of the root-face, are also extracted.