COUNTING ROOTED EULERIAN MAPS ON THE PROJECTIVE PLANE

1IntroductionAsurfaceisacompactclosed2-manifold.Theorielltable(non-orielltable)surfaceofgenuskisthespherewitllkhandles(crosscaPs)denotedbySk(Nk).AmapMollSk(Nk)meansthatitsunderlyinggraphnlaybedrownou(embeddedin)itsuchthatllthpairofedgesintersectataninnerpoilltalldeachfaceishomeomorphictothedisc.Amapisrootedifanedgewithadirectiollalongtheedge,alldasideoftl1eedgeisdistinguisl1ed.Tworootedmapsareconsideredtobethesal11eifthereisanisomorphismpreserviIlgtl1erooting.ArootedEuleriall1llapissuchaon…     

某些曲面上有根近-4-正则欧拉迹的计数

In this article the rooted planar near-4-regular Eulerian trails are enumerated and an explicit formula for such maps is presented. Further, the rooted near-4-regular Eulerian maps on the torus are counted in an exact way.     

柱面上有根近三角剖分的计数

柱面上的三角剖分是一类与环面上的地图紧密相关的地图．本文提供了一个计算柱面上有根近三角剖分的具有三个变量的精确公式．     

Counting rooted near-4-regular Eulerian maps on some surfaces

In this article the rooted planar near-4-regular Eulerian trails are enumerated and an explicit formula for such maps is presented. Further, the rooted near-4-regular Eulerian maps on the torus are counted in an exact way.     

极大平面图的组合运算

一、引言有关平面4色问题的综合论述见文[1]和文[2].目前这一问题因有计算机辅助证明而得到解决,但这种证明繁杂.又由于等价的命题多,联系的方面广,因而对此问题进行新的理论探讨,便不会是毫无意义.把平面4色问题化成只与圈上的4染色集性质有关的问题来解决,这种思想在文[3]中已经有了,只可惜该文最后所提出的猜想不真(对长度为4的圈已不成立.此事我们将另文评述).本文以极大平面图的组合序列为基础,得出一些只与圈上的4染色集的性质有关的等价于平面4色猜想的命题.     

On the Number of Fair Triangulations

In this paper a special kind of triangulated maps on the sphere called fair triangulations is enumerated with the size of maps as parameter. Moreover, the number of several other kinds of triangulations are enumerated as well. Received June 18, 1997, Accepted June 12, 1998     

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。     

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 。     

BISINGULAR MAPS ON SOME SURFACES

A map is bisingular if each edge is either a loop (This paper only considersplanar loop) or an isthmus (i.e., on the boundary of the same face). This paper studies thenumber of rooted bisingular maps on the sphere and the torus, and also presents formulaefor such maps with three parameters: the root-valency, the number of isthmus, and thenumber of planar loops.     

环面上有根欧拉地图的计数

本文探讨了环面上有限欧拉迹的计算并且提供了一个解析表达式．在此基础上，我们给出了环面上有根欧地图的计算公式．     

The number of rooted essential maps on surfaces

In this paper, general functional equations of rooted essential maps on surfaces （orientable and nonorientable） are deduced and their formal solutions are presented. Further, three explicit for- mulae for counting essential maps on S2,N3 andN4 are given. In the same time, some known results can be 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…     

Counting Non separable Near triangulations on the Mbius Band

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关于无穷级半纯函数的填充圆

本文应用Ａｈｌｆｏｒｓ覆盖曲面理论，在一定条件下证明了无穷级半纯函数强性填充圆的存在性，从而部分地解决了李国平提出的一个猜想。     

匀称无圈超图的计数

研究了标号匀称无圈超图的计数, 得到了一般的$n$阶标号r-匀称(d)-森林和n阶标号r-匀称(d)-真森林的递推公式,并分别得到了包含和不包含独立点的$n$阶标号森林的计数显式.     

ENUMERATIONS OF ONE-VERTEX MAPS WITH FACE PARTITION ON THE PLANE

In addition to the known method given in [1],authors provide other three methods to the enumeration of one-vertex maps with face partition on the plane.Correspondingly,there are four functional equations in the enufuntion .It is shown that the four equations are equivalent.Moreover,an explicit expression of the solution is found by expanding the powers of the matrix of infinite order directly.This is a new complement of what appeared in [1].     

Radon变换和衰减Radon变换的分析研究

衰减Radon变换出现在单光子放射型计算机层析成像中。本文首先回顾和研究了Radon变换和衰减Radon变换及其反演的有关结论，进而提出了Tretiak-Metz结果的一种新证明方法，对于一般对象，本文用变换方法非滤子背投影法导出了衰减Radon变换的反演公式。