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.     

Chromatic sums of biloopless nonseparable near-triangulations on the projective plane

In this paper, the chromatic sum functions of rooted biloopless nonseparable near-triangulations on the sphere and the projective plane are studied. The chromatic sum function equations of such maps are obtained. From the chromatic sum equations of such maps, the enumerating function equations of such maps are derived. An asymptotic evaluation and some explicit expression of enumerating functions are also derived.     

Chromatic sums of nonseparable near-triangulations on the projective plane

In this paper, we study the chromatic sum functions of rooted nonseparable near-triangulations on the sphere and the projective plane. The chromatic sum function equations of such maps are obtained. From the chromatic sum equations of such maps, the enumerating function equations of such maps are derived. Applying chromatic sum theory, the enumerating problem of different sorts maps can be studied, and a new method of enumeration can be obtained. Moreover, an asymptotic evaluation and some explicit expression of enumerating functions are also derived.     

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。     

Chromatic sums of 2-edge-connected maps on the projective plane

This paper provides the chromatic sum function equations of rooted 2-edge-connected maps on the projective plane. The enumerating function equations of rooted 2-edge-connected loopless maps and rooted 2-edge-connected bipartite maps on the projective plane are derived by the chromatic sum function equation of rooted 2-edge-connected maps on the projective plane.     

Chromatic sums of general maps on the sphere and the projective plane

In this paper, we study the chromatic sum functions of rooted general maps on the sphere and the projective plane. The chromatic sum function equations of such maps are obtained. From the chromatic sum equations of such maps, the enumerating function equations of rooted loopless maps, bipartite maps and Eulerian maps are also derived. Moreover, some explicit expressions of enumerating functions are also derived.     

Chromatic sums of rooted triangulations on the projective plane

In this paper we study the chromatic sum functions for rooted nonseparable near-triangular maps on the projective plane. A chromatic sum equation for such maps is obtained.     

射影平面上的对偶无环不可分近三角剖分地图

本文研究了球面和射影平面上对偶无环不可分近三角剖分带根地图的以根面次和内面数为参数的计数问题,得到了这类地图在球面和射影平面上的计数函数满足的方程.还得到了射影平面上2连通地图一个参数的显示表达式和渐近估计式.     

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.     

Chromatic Enumeration for Rooted Outerplanar Maps Liu Yanfiei

Let G(m, s, t; \lambda)be the number of ways of \lambda-coloring all the rooted nonseparable outerplanar maps which are simple and have the edge number m, the valency s of the root-face, and the valency t of the root-vertex. The chromatic enumerating, function $g(x,y,z;\lambda)=\sum\limits_{m\geq 1,s\geq 2,t\geq 2}{G(m,s,t;\lambda)x^my^sz^t$ is determined. Meanwhile, a number of explicit formulae for enumerating this kindof maps in general case and in bipartite ease are provided.     

关于平面树色和方程结果的一个注记

这篇文章得到了有根平面树的节点剖分的色和方程. 导出了带无限多个参数的有根平面植树和平面树的色和方程的精确表达式. 作为直接推论可推出节点剖分的有根平面树的计数方程的精确结果 .     

Chromatic sums of singular maps on some surfaces

A map is singular if each edge is on the same face on a sruface (i.e., those have only one face on a surface). Because any map with loop is not colorable, all maps here are assumed to be loopless. In this paper povides the explicit expression of chromatic sum functions for rooted singular maps on the projective plane, the torus and the Klein bottle. From the explicit expression of chromatic sum functions of such maps, the explicit expression of enum erating functions of such maps are also derived.     

Enumeration of loopless maps on the projective plane

In this paper we study the rooted loopless 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 function is 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.     

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…     

带根2-连通3-正则平面地图计数

本文利用不可分离的3-正则有根平面地图的计数结果,间接地给出了2-连通 3-正则有根平面地图依边数和根面次的计数显式．     

The number of rooted nearly cubicC-nets

This paper provides the parametric expressions satisfied by the enumerating functions for rooted nearly cubicc-nets with the size and/or the root-vertex valency of the maps as the parameters via nonseparable nearly cubic maps. On this basis, two explicit expressions of the functions can be derived by employing Lagrangian inversion. This Research is supported by National Natural Science Foundation of China (No. 19831080).     

On the vertex partition equation of loopless Eulerian planar maps

The functional equation satisfied by the vertex partition function of rooted loopless Eulerian planar maps is provided. As applications, the enumerating equations for general and regular cases of this kind of maps are also discussed.This project is supported partially by the National Natural Science Foundation of China Grant 18971061.     

环面上一般有根地图的计数

这篇文章给出了环面上以内面个数，根面次和非根节点个数为参数的一般有根地图的计数方程，导出了以内面个数和非根节点个数为参数的这类地图的计数方程的精确解。作为推论，推出了以边数为参数的这类地图的个数，其近似解在文献[2]中已讨论。     

ENUMERATION OF ROOTED VERTEX NON-SEPARABLE PLANAR MAPS

In a rooted planar map, the rooted vertex is said to be non-separable if the vertex onthe boundary of the outer face as an induced graph is not a cut-vertex. In this paper, the author derives a functional equation satisfied by the enumeratingfuuction of rooted vertex non-separable planar maps dependent on the edge number and thenumber of the edges on the outer face boundary, finds a parametric expression of itssolution, and obtains an explicit formula for the function. Particularly, the number of rooted vertex non-separable maps only replying on theedge number and that of rooted vertex non-separable tree-like maps defined in [4] accordingto the two indices, the edge number and the number of the edges on the outer face boundary,or only one index, the edge number, are also determined.     

DICHROMATIC SUM EQUATIONS FOR OUTERPLANAR MAPS

This paper provides a functional equation satisfied by the dichromatic sum funetion of rooted outerplanar maps.By the equation,the dichromatic sum funetion can be found explicitiy.