Research on Combinatorial Geometry

组合几何研究

基本信息

批准号：
11640135
负责人：
OTA Katsuhiro
金额：
$ 2.24万
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：
1999
资助国家：
日本
起止时间：
1999 至 2000
项目状态：
已结题

项目摘要

Triangulations and Quadrangulations of a closed surface are most typical combinatorial objects related with geometic ones. We first consider the problem to transform one triangulation (or quadrangulation) to another by a sequence of local deformations. Continued to the preceding research, we have obtained some results for degree constrained cases and for outer-triangulations, a generalization of outer planar graphs. We have also obtained an interesting result on planar triangulations which can be embedded as a quadrangulation of another surface. In particular, it depends on the orientability of the surface, and certain combinatorial structure of the triangulation plays a key role.Coloring of quadrangulation has a similar phenomenon that appears in only nonorientable surfaces. If the quadrangulation has a cycle cutting the surface into orientable one, then the chromatic number is at least 4. In particular, the chromatic number of a quadrangulation of torus and Klein bottle is determined by a topological and algebraic invariant of the graph.High representativity of a 3-connected graph G on a surface enable us to cut open G into a suitable plane graph. We have established a very useful tool describing such a cutting. Using this tool, we have proved several theorems that show 3-connected graphs on a surface are close to be hamiltonian in a sense ; concerning spanning tree with maximum degree at most four, spanning 2-connected subgraph with maximum degree at most eight, and light connected subgraphs with given size.

封闭表面的三角剖分和四边形是与地貌相关的最典型组合物体。我们首先考虑通过一系列局部变形将一个三角剖分（或四角）转换为另一个三角剖分的问题。继续进行上一项研究，我们为程度受限的病例和外部习惯（外部平面图的概括）获得了一些结果。我们还获得了平面三角剖分的有趣结果，该结果可以嵌入另一个表面的四边形。特别是，这取决于表面的可方向性，三角剖分的某些组合结构起着关键作用。四角形的颜色具有相似的现象，仅在不可定向的表面出现。如果四角形的周期将表面切成可定向的周期，那么色数至少为4个。特别是，圆环和klein瓶的四个四边形的色数由图形的拓扑和代数不变性确定。高度代表性的高度代表性，使表面上的表面上的G g of apphs of Applappie of Us opape of phope opape op opape op opaper opape opape opape opape opape opape opape opation gopth opation gope opain。我们已经建立了一个非常有用的工具来描述这种切割。使用此工具，我们已经证明了几个在表面上显示3个连接图的定理，从某种意义上说是哈密顿量。关于最高度最高度的跨越树，最多跨越了2个连接的子图，最多八个，以及带有给定尺寸的光连接子图。