On Nonisomorphic Maps Of Given Genus

The purpose of this thesis is to discuss the nonisomorphic rooted maps of given genus, that is, the enumeration of rooted maps which is the basis of the enumeration of maps. It is divided into two main parts: enumeration of planar maps and enumeration of nonplanar maps. A general method for counting rooted maps is by employing analytic technique, which plays an important role in the enumeration of maps. In addition, in this thesis, some skills of combinatorics are also embodied.For studying the nonisomorphic rooted maps on the sphere, an interesting important step in our method involves giving an appropriate decomposition for the set of maps considered. The content of this part is concentrated on the first three chapters.In chapter 1, firstly, we give a brief introduction about the background of the enumerative theory of maps. Then some definitions and properties, relate to maps, are provided. Finally, the analytic technique is expounded in detail.Chapter 2, dicusses the planar pan-fan maps and circuit boundary maps, which brought forward based on the Halin maps. The importance of Halin maps has been discovered in graph embedding theory as well as in the enumerations of maps. These two kinds of maps are more general than Halin maps. In this chapter, using two different methods, we obtain their explicit expressions with different parameters. Meanwhile, some explicit expressions about the loopless pan-fan maps, circuit cubic boundary maps and strict circuit boundary maps are derived.In chapter 3, we study the out-chordless general maps which derive from nonseparable maps. Out-chordless maps were proposed by Y.P.Liu in lit. [51] when he enumerated cubic maps on the projective plane. In lit. [24], R.X.Hao obtained the number of out-chordless nearly cubic planar maps. Here, by searching the relation of the out-chordless maps and the nonseparable planar maps, we get the number of out-chordless general maps with the size as a parameter.On the basis of the enumerative theory of rooted planar maps, the second part which includes chapter 4, 5 and 6, mainly investigate the number of nonisomorphic maps on the surfaces with genus g > 0. The crux of this problem is to choose suitable parameters, since here the functional equations are always " complex ".In chapter 4, petal bundles on surfaces are studied detailedly. As we know, the dual maps of all petal bundles on the surface of genus g are singular maps (or g-essential maps). These two kinds of maps are of the simplest class. A particular discussion about these maps can lay a foundation on our further works, especially, on surfaces with higher genus. A lot of scholars studied them. However, exact enumeration was mainly centralized on surfaces with small genus, or the results were complex. In this chapter, we provide the uniform enumerative functional equation of orientable(nonorientable) rooted petal bundles and deduce two recursion formulas for calculation. Accordingly, the number of these maps with up to two parameters on the orientable surfaces(genus 0-3) and the nonorientable sur-faces(genus 1-5) are obtained. As corol., some explicit expressions of rooted petal bundles on some surfaces with the size n are also gained.Based on the forth chapter, in the fifth chapter, we consider a kind of maps: nearly 2-regular maps, topologically equivalent to petal bundles. By a combinatorial method, the explicit expression of rooted nearly 2-regular maps on the sphere with three parameters is provided. And a relation between these maps and petal bundles on surfaces with arbitrary genus is also found.In chapter 6, we obtain a general enumerating functional equation about rooted pan-fan maps on nonorientable surfaces. Basen on this equation, two explicit expressions of rooted pan-fan maps on the projective plane and the Klein bottle are given.Chapter 7, discusses some unresolved problems about the enumeration of maps. Such as, exact enumeration of maps on the higher genus; enumeration of maps by probabilistic methods etc.