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.     

射影平面上不可分近三角剖分地图的计算

众所周知,由于本质圈(或不可收缩圈)的作用,使得一般的曲面上要得到带有两到三个参数的地图计算公式(尤其是显式公式)变得十分困难。该文集中讨论射影平面上不可分近三角剖分地图的计算。通过引入含有面次,边数和内部面数的参数表达式与Lagrangian反演,作者得到了含有正项系数的显式公式用以计算射影平面上三角剖分地图 。     

射影平面和环面上奇异地图的色和

一个地图的每条边,若在同一面的边界上,则称它为奇异地图．由于含环的地图是不可着色的,本文所有地图均不含环．本文研究射影平面和环面上带根奇异地图的色和．     

一类循环图在射影平面上的嵌入北大核心CSCD

嵌入的联树模型是研究图的曲面嵌入的一种有效方法,尤其能方便快捷地研究图在球面,环面,射影平面,Klein瓶上的嵌入。此方法通过合理选择生成树,得到联树和关联曲面,然后对关联曲面进行计数,计算出图在曲面上的嵌入个数.本文利用嵌入的联树模型得出了循环图C(2n+1,2)(n>2)在射影平面上的嵌入个数.     

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。     

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.     

关于有根简单平面三角地图数目的一个注记

一个地图,指图在某曲面上的一个嵌入.平面地图,自然就是在平面上的嵌入.三角平面地图,即所有面皆三角形的平面地图.一个地图,如果将它的一条边规定一个特殊的方向,此边称为根边.根边的始端称为根点,沿根边的方向走左手边的面称为根面,则这时称这个地图为有根的.之谓二个有根的地图是不同的,或曰组合上不等价,指二者之间不存在一个同构映像使得它们根点、根边和根面也相应.可想而知,研究有根地图的组合不变性与研究一般地图的组合不变性在根多情况下是一致的.特别是在着色理论中有根和无根一个样.然,     

Klein瓶上的双奇异地图

一个地图的每条边如果不是环就是割边(即该边的两边是同一个面的边界)，则称之为双奇异地图，本文研究Klein瓶上带根双奇异地图的计数问题，得到了此类地图以边数、平面环数、手柄上本质环数和又帽上本质环数为参数的计数公式，并得到了部分计数显式。     

循环图C(2n,2)(n＞2)在射影平面上的嵌入计数

图在球面上的嵌入个数即柔性问题已经由刘彦佩教授解决,研究图在射影平面上的嵌入亦有着重要的意义。本文利用刘彦佩教授创建的嵌入联树模型得出了循环图C(2n,2)(n>2)在射影平面上的嵌入个数。     

基于有限射影平面上(ω,r,d)-CFF(N,T)系统的构作

在有限射影平面上利用有限射影平面的性质构作了(ω,r,d)-CFF(N,T)系统,并利用有限射影平面的性质计算了它的参数.最后利用一个有限点集构作了一个(ω,r,d)-DS(N,T)系统并计算了它的参数.     

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 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 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 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 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.     

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.     

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.     

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.