权益分类	功能权益	普通用户	{{item.name}}会员
{{category.name}}	{{benefitItem.name}}

Research on Geometric and Combinatorial Structures

几何与组合结构研究

基本信息

批准号：
09640290
负责人：
OTA Katsuhiro
金额：
$ 2.37万
依托单位：
Keio University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1997
资助国家：
日本
起止时间：
1997 至 1998
项目状态：
已结题

项目摘要

Triangulations and Quadrangulations of a closed surface with sufficiently fine mesh will approximate the discrete structure of the surface. We first consider the problem to transform one triangulation (or quadrangulation) to another by a sequence of local deformations. As a result for triangulations, if these graphs have same number of vertices and the number is large enough, namely, these graphs are good approximation of the surfacs, then these can be transformed each other by a sequence of diagonal flips. On the other hand, for quadrangulations, there exists an discrete invariant "cycle parity" that is closely related to the structure of the mapping class group of the surface.In a graph embedded in a closed surface, the degree of a vertex is considered as a discrete curvature on that place. In a 3-connected graph on the sphere, we studied the minimum degree sum of a connected subgraph having prescribed order.We also consider the problem of embedding graphs in a book-type manifold. We gave good bounds for the pagenumber of complete bipartite graphs, where the pagenumber of a graph is the minimum number of pages in which one can embed the graph without crossing edges and the spine. For general graphs, we gave a lower bound for the number of crossing points of edges and the spine.Also, we investigate the following topics ; a presentation of mapping class groups in terms of Artin groups, Galois action on fundamental groups, study of difference sets using number theory, searching irreducible polynomials, isometric embedding of simplexes. and the problem to represent an arbitrary graph as an integral distance graph in the plane.

三角剖分和四边形剖分的一个封闭的表面与足够细的网格将近似的离散结构的表面。首先考虑通过一系列局部变形将一个三角网（或四边网）变换为另一个三角网的问题。作为三角剖分的结果，如果这些图具有相同数量的顶点并且数量足够大，即这些图是曲面的良好近似，则这些图可以通过一系列对角翻转相互转换。另一方面，对于四边形剖分，存在一个离散不变的“圈宇称”，它与曲面的映射类群的结构密切相关。在嵌入闭曲面的图中，顶点的度被认为是该位置上的离散曲率。在球面上的3-连通图中，研究了具有给定阶的连通子图的最小度和问题，并考虑了图在书型流形中的嵌入问题。我们给出了完全二部图的页数的一个好的界，其中图的页数是可以嵌入图而不交叉边和脊的最小页数。对于一般图，我们给出了边与脊交叉点个数的下界，并研究了以下问题：用Artin群表示映射类群，Galois作用在基本群上，用数论研究差集，寻找不可约多项式，单形的等距嵌入.以及将任意图表示为平面上的整数距离图的问题。