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.     

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.     

Counting rooted 4-regular unicursal planar maps

A map is 4-regular unicursal if all its vertices are 4-valent except two odd-valent vertices. This paper investigates the number of rooted 4-regular unicursal planar maps and presents some formulae for such maps with four parameters: the number of edges, the number of inner faces and the valencies of the two odd vertices.     

数有根近2-正则平面地图

The number of rooted nearly 2-regular maps with the valency of root-vertex, the number of non-rooted vertices and the valency of root-face as three parameters is obtained. Furthermore, the explicit expressions of the special cases including loopless nearly 2-regular maps and simple nearly 2-regular maps in terms of the above three parameters are derived.     

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。     

Bisingular maps on the torus

A map is bisingular if each edge is either a loop or an isthmus (i.e., on the boundary of the same face). In this paper we study the number of rooted bisingular maps on the sphere and the torus, and we also present formula for such maps with four parameters: the root-valency,the number of isthmus, the number of planar loops and the number of essential loops.     

关于平面上双树梵和的一个注记

这篇文章得到了以根节点的次、割边的个数及环的个数为参数的双树梵和的色和方程，且导出了这类地图带以上三个参数的精确解及一些退化的情形。     

Enumeration of r-regular maps on the torus. Part I: Rooted maps on the torus,the projective plane and the Klein bottle. Sensed maps on the torus

The work that consists of two parts is devoted to the problem of enumerating unrooted $r$-regular maps on the torus up to all its symmetries. We begin with enumerating near-$r$-regular rooted maps on the torus, the projective plane and the Klein bottle, as well as some special kinds of maps on the sphere: near-$r$-regular maps, maps with multiple leaves and maps with multiple root darts. For $r=3$ and $r=4$ we obtain exact analytical formulas. For larger $r$ we derive recurrence relations. Then we enumerate $r$-regular maps on the torus up to homeomorphisms that preserve its orientation — so-called sensed maps. Using the concept of a quotient map on an orbifold we reduce this problem to enumeration of certain above-mentioned classes of rooted maps. For $r=3$ and $r=4$ we obtain closed-form expressions for the numbers of $r$-regular sensed maps by edges. All these results will be used in the second part of the work to enumerate $r$-regular maps on the torus up to all homeomorphisms — so-called unsensed maps.     

Counting unrooted maps on the plane

A planar map is a 2-cell embedding of a connected planar graph, loops and parallel edges allowed, on the sphere. A plane map is a planar map with a distinguished outside (“infinite”) face. An unrooted map is an equivalence class of maps under orientation-preserving homeomorphism, and a rooted map is a map with a distinguished oriented edge. Previously we obtained formulae for the number of unrooted planar n-edge maps of various classes, including all maps, non-separable maps, eulerian maps and loopless maps. In this article, using the same technique we obtain closed formulae for counting unrooted plane maps of all these classes and their duals. The corresponding formulae for rooted maps are known to be all sum-free; the formulae that we obtain for unrooted maps contain only a sum over the divisors of n. We count also unrooted two-vertex plane maps.     

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.     

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.     

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．     

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.     

Counting labeled general cubic graphs

Recurrence relations are derived for the numbers of labeled 3-regular graphs with given connectivity, order, number of double edges, and number of loops. This work builds on methods previously developed by Read, Wormald, Palmer, and Robinson.     

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。     

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.     

Enumeration of rooted planar halin maps

A Halin map is a kind of planar maps oriented by a tree. In this paper the rooted halin maps with the vertex partition as parameters are enumerated such that a famous result on rooted trees due to Harary. Prins, and Tutte is deduced as. a special ease. Further, by using Lagrangian inversion to obtain a number of summation free formulae dixectly, the various kinds of rooted Halin maps with up to three parameters have been counted.     

Enumerative Properties of Rooted Circuit Maps

In 1966, Barnette introduced a set of graphs, called circuit graphs, which are obtained from 3-connected planar graphs by deleting a vertex. Circuit graphs and 3-connected planar graphs share many interesting properties which are not satisfied by general 2-connected planar graphs. Circuit graphs have nice closure properties which make them easier to deal with than 3-connected planar graphs for studying some graph-theoretic properties. In this paper, we study some enumerative properties of circuit graphs. For enumeration purpose, we define rooted circuit maps and compare the number of rooted circuit maps with those of rooted 2-connected planar maps and rooted 3-connected planar maps.     

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.     

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.