Combinatorial Methods for Discrete Geometry

In this project，我们首先extracted the combinatorial aspects from a number of problems in discrete geometry，和分类他们。Then by invetigating each category，我们tried to establish general methods which are applicable to discrete geometry. The following aresome of the most successful results in this project. in discrete geometry，there are a number of problems which are essentially equivalent to joining points by straight linesegments so that the resulting geometric object becomes a hamiltonian cycle embedded in the plane这是line segments的第一个数字，我们提倡他们就是组合problems in essence，我们提倡独立的问题，我们提倡的combinatorial方法geometric trees可以handled as an extension of independent trees in graphs，我们establlished a graph-theoretic approach to address these problems. In discrete geomtery，there are many problems on dividing an Euclidean space with a finite number of geometric objects bya hyperplane so that both divided half-spaces contain almost the same number of the objects. Westudied the combinatorial aspects of these problems在其中每个对象都是一个球的问题，我们提供了许多问题discrete geometry which are essentially graph decomposition problems.我们解决了一个数字decomposition problems,especially in case of complete graphs and complete bipartite graphs. the above results are just asmall fraction of our entire resultswhich are described fully in the project report. Considering the quantity of the quality of the结果，我们believe that this project是extremely successful。

In this project, we first extracted the combinatorial aspects from a number of problems in discrete geometry, and categorized them. Then by invetigating each category, we tried to establish general methods which are applicable to discrete geometry. The following are some of the most successful results in this project.・ In discrete geometry, there are a number of problems which are essentially equivalent to joining points by straight line segments so that the resulting geometric object becomes a hamiltonian cycle embedded in the plane and that it has the least number of crossing of line segments. We proved that they are combinatorial problems in essence, and established a combinatorial method to tackle them.・ We proved that the problems of independent geometric trees can be handled as an extension of independent trees in graphs, and we establlished a graph-theoretic approach to address these problems.・ In discrete geomtery, there are many problems on dividing an Euclidean space with a finite number of geometric objects by a hyperplane so that both divided half-spaces contain almost the same number of the objects. We studied the combinatorial aspects of these problems, and solved the problem in which each object is a ball.・ We proved that there are many problems in discrete geometry which are essentially graph decomposition problems. We solved a number of these decomposition problems, especially in case of complete graphs and complete bipartite graphs.The above results are just a small fraction of our entire results, which are described fully in the project report. Considering the quantity of the quality of the result, we believe that this project was extremely successful.