具有缩影W_8和W_9的图式流形

Graphlike Manifolds with Contraction W_8 and W_9

图式流形是将简单无向图中的所有边用管取代、所有顶点用圆周替换而得到的一种新管型曲面。这些管型曲面即是图式流形,而简单无向图则称为相应图式流形的缩影。若图式流形的圆周采取不同的覆盖映射,则可得到不同的图式流形。这些图式流形有无限多个,而计算所有这些图式流形同胚分类的个数,并给每一种同胚类型指定一个图式流形的代表,即为图式流形的拓扑分类问题。本文从图染色理论出发,对收缩为n个顶点的轮图Wn进行了研究,探讨收缩为Wn的图式流形同胚等价类的个数,以及所有互不同构的着色构成代表系需要满足的条件。利用图论中的边染色理论结合扭转运算,在同胚的意义下得到并绘出了具有收缩W8、W9的图式流形的代表图形,它们分别只有18和30个。

Consider an undirected simple connected graph, in which each vertex is replaced by a manifold （a pipe） and each edge is replaced by the Cartesian product of this manifold （a circlar section）. The topological space obtained in this way is called a graphlike manifold. The undirected simple graph is called the contraction of graphlike manifolds correspondingly. If the circlar sections of graphlike manifolds have different maps, we will have different graphlike manifolds. There are infinite graphlike manifolds, and the number of homeomorphic classes is also hard to count, nearly infinity. The problem of counting the numbers of homeomorphics classes of graphlike manifolds, and give each homeomorphics classes a representive graphlike manifold, is just the case for the topological classification of graphlike manifolds. This paper discusses the topological classification of graphlike manifolds with the contraction of Wn and the number of homeomorphic classes of graphlike manifolds of Wn. A representative system is formed by all non-isomorphic colorings, and the necessary cases are counted. According to the graph colouring theory and twist operation, the homeomorphic classes of grapglike manifolds Ws and W9 are found to be 18 and 30, respectively.