An Asymmetric Census Of Maps On Surfaces

Starting with W. T. Tutte's seminal works on the planar triangulation in the beginning of 1960s, enumerative theory of maps has been studied intensively. In the past of over fifty years the theory was developed and enriched step by step. Now, enumerative theory of maps includes the census of maps, the chromatic sum, dichromatic sum of maps and so on. This thesis mainly focuses on an asymmetric census of maps on surfaces. Both planar maps and non-planar maps are investigated and some new results are obtained.In Chapter 1, it is briefly introduced that the background and significance of the problem considered, some basic definitions and the technologies employed in the thesis. Then the skeleton of the thesis is described.Chapter 2 is mainly concerned with planar maps. It is well known that planar Halin maps are so important that they are widely applied in many fields. In this chapter, boundary cubic rooted planar maps, as the extensions of planar Halin maps, are investigated and exact enumerative formulae are given. First, an enumerative formula for boundary cubic inner-forest maps with the size as a parameter is derived. Then, for the special case of boundary cubic inner-tree maps, a simple formula with two parameters is presented. Further, according to the duality, a corresponding result for outer-planar maps is obtained. Finally, some results for boundary cubic planar maps arc obtained. Furthermore, two known Tutte's formulae arc easily deduced in the chapter.We discuss non-planar maps in Chapter 3. On the enumeration of maps on the nonorientable surfaces, most of literatures concentrated on such surfaces as the projective plane and the Klein bottle. In 2000, D. Arques [3] first presented the formulae for counting maps on the nonorientable surfaces with genus 3 and 4. In this chapter, we investigate essential maps on surfaces which play important roles in the study of gen- eral maps on surfaces. First, exact enumerating formulae of 2-essential maps on N₂, 3-essential maps on N₃ and 4-essential maps on N₄ are derived. Then, the formula for counting 5-essential maps on N₅ is presented, which is the first enumerative result of maps on N₅. In the same time, the corresponding result for counting 2-essential maps on S₂ is also obtained. Finally, general functional equations of essential maps on two kinds of surfaces(orientablc and nonorientable) are deduced and their formal solutions are presented.In Chapter 4, rooted general maps on all surfaces (orientable and nonorientable) are considered. Liu[67] had obtained the enumerating equation with the edge as a parameter for rooted general maps on all surfaces before. Basing on it, rooted general maps on all surfaces are further investigated and the enumerating equation with respect to vertices and edges is derived, which is a Riccati's equation. Now, not only the analytical solution but also the series solution of Riccati's equation is hard to be obtained . To solving it, a new solution in continued fraction form is given. As two especial cases, the corresponding results of rooted general maps and monopole maps on surfaces with respect to edges are obtained, which answer an open problem [67].The last chapter summarizes the thesis and plans to the further works, which are the following four problems:Problem 1: to count essential maps on the nonorientable surfaces with arbitrary genus;Problem 2: to count general maps on the nonorientable surfaces with arbitrary genus;Problem 3: to count unrooted maps on surfaces;Problem 4: the series solution of the Riccati's equation.