平面图

Specially,we construct some new kinds of planar graphs with circular chromatic number 3 or 4 based on the results which have been surveyed in part two.

其中第二部分在总结平面图的圈色数已有结论的基础上，还构造了一些新的圈色数为3或4的平面图。

The main work of this paper is that we have given the edge game chromatic number of cycle likes, fan graphs, flowers, Halin graphs, and the upbound is sharp for cycle likes and Halin graphs, it is also a good example for the study of planar and outerplanar graphs.

主要的工作是对一些特殊图类的边对策色数进行了研究，给出了类圈图，扇图，花形图和Halin图的边对策色数的上界，并且证明了关于类圈图和Halin图的结果是紧的，这就给平面图，外平面图的研究提供了很好的例子。

Let G be a double-outer plane graph, V, E, F be the set of points, edge and face, a plane graph G is k-total colorable, if the elements of V E can be colored with K-colors, any two distinct adjacent or incident elements receive different colors.

设G是一个双外平面图，V，E，F分别为双外平面图G的点集，边集和面集。G的全色数X是使得V∪E中的任意两个相邻或相关联的元素间均染不同颜色的最少颜色数。

Then we prove the one-to-one corresponding relation between 3-coloring of inducing four regular graph and 4-coloring of maximal planar graph and find out the relation between three colors of inducing four regular graph and three dual bi-chromatic subgraph of maximal planar graph.

本文给出了极大平面图的导出四正则图的两种构造方式、等价性及性质，证明了导出四正则图的三着色与原极大平面图四着色的一一对应关系，并且找出了导出四正则图的三种颜色与原极大平面图四着色的三组对偶二色子图之间的关系。

In section four, by researching the properties of Kneser graphs, we construct the homomorphism φ between the Kneser graph and the outerplane graph. So the fractional chromatic number and chromatic number are obtained.

在第四部分中通过对Kneser图的一些性质的考察，在Kneser图与外平面图之间建立同态关系，从而得到了外平面图的分数色数与顶点色数。

In noncommutative scalar field theories, the appearing of nonplanar diagrams is related to permutation symmetry of field functions in the interaction Lagrangian, while in NCQED we can not arbitrarily change the positions of field functions in the interaction Lagrangian because different positions of the field functions correspond to different representations of the gauge group.

这里需要特别指出的是NCQED的非平面图产生的原因与非对易标量场根本不同，非对易标量场非平面图的产生是和相互作用拉氏量密度中场函数间位置的置换对称性有关，而在NCQED中我们不能随便交换相互作用拉氏量密度中场函数的位置，因为场函数的不同位置对应着规范群的不同表示。

The following are included in the system: GIS client editing tools; use of the VBA editing environment in the ARCINFO platform; functions custom designed to meet customer needs; and use of the ARCSDE and ARCIMS tools in ARCGIS to manage data space in the E-map and provide to the internet. We have developed a module, using ichnography that is used in the system browser to manage microwave station information visually.

系统中与图形有关的设计和实现包括：GIS客户化编辑工具，我们主要采用了ARCINFO自带的VBA编辑环境，重新按照客户需求定制了自己的功能；使用ARCGIS中的ARCSDE和ARCIMS提供的工具对电子地图空间数据管理和互联网上发布；系统微波站图形方式浏览和管理，我们自己开发了一个模块，利用平面图作为页面的母层，平面图上需要表现的设备、维护人员等传输资源建立子层链接，通过链接可以浏览详细信息。

In the same way, a Four-coloring of a maximal planar graph of 25 order G(subscript M25), is obtained with "simplified method of reduction of order".

利用"简化降阶法"对一个一定拓扑结构的12阶最大平面图G(下标 M12)进行着色，得到G(下标 M12)的四色着色方案；以同样的方法对一个一定拓扑结构的25阶最大平面图G(下标 M25)进行着色，得到了G(下标 M25)的四色着色方案。

The method defines two types of oriented 4-valent plane graph: G_e and G_o, whose vertices are covered by E-tangles and O-tangles, respectively.

这一方法定义了两种有向4-度平面图：G_e和G_o分别用E-tangles和O-tangles覆盖两种有向4-度平面图的顶点。

Some sufficient conditions for a planar graph to be class 1 are obtained through these bounds of edge number.

给出了平面图为第一类图的边数的一些上界，并给出了平面图为第一类图的一些充分条件。

More than five decades after the Indian constitution promises a casteless society, there is no sign that the castes are disappearing.

在印度宪法允诺一个&无种姓&社会五十多年后，种姓尚没有显示出任何要消失的迹象，但变化也确确实实地发生了。

Don't distract him from his studies.

他在学习不要打扰他。