平面图的

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。

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 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。

Used the way of discharging to prove that every planar graph without 4-cycles and i-cycles for some i ∈{5,6,7} is (3, l)*-choosable, when they studied the list improper colorings of plane graphs.

72在研究平面图的列表非正常染色时，结合欧拉公式用放电的方法证明了每个不含4-圈和i-圈的平面图是(3，1)~*-可选的，其中i∈{5，6，7}。

In section four,we will study the star-edge coloring of maximal outer plane graph and maximal plane graph,and bound their star chromatic index.

第四节研究了极大外平面图和极大平面图的星边染色，并界定了它们的星边色数。

We also get another expression of Steinberg's conjecture about plane graphs, that is, every plane graph without i(4≤i≤5) faces in which any two triangles have distance at least 3 and no 3-face is adjacent to any 6-faces is 3-colorable.

我们还得到平面图的一个类似结论：没有4、5面的平面图中如果任意两个3圈的距离至少为3，而且没有3面和6面相邻，那么这个图也是可以3色的。

We also get another expression of Steinbergs conjecture about plane graphs, that is, every plane graph without i(4≤i≤5) faces in which any two triangles have distance at least 3 and no 3-face is adjacent to any 6-faces is 3-colorable.

我们还得到平面图的一个类似结论：没有4、5面的平面图中如果任意两个3圈的距离至少为3，而且没有3面和6面相邻，那么这个图也是可以3色的。

This is not an ordinary box, which is used as a background picture of the dialog box, very pretty.

详细说明：这可不是一个一般的对话框，它是用图片作为背景的对话框，非常好看。

Conceal me what I am,and be my aid for such disguise as haply shall become the form of my intent.

遮掩我的身份，帮助我，我的面具将成为我的目的。