### 数学代写|图论作业代写Graph Theory代考| For each graph below

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|图论作业代写Graph Theory代考|EXERCISES

1. For each graph below and on the following page, first apply the exclusion test using Euler’s formula to determine one of the following:
• The graph is definitely nonplanar.
• The graph could be a MPG.
• The graph could be a PG, but not a MPG.
Now determine if each graph is a MPG, PG but not a MPG, or nonplanar. If a graph is a MPG or PG, redraw the graph to show that the graph is a PG. If a graph is nonplanar, either apply Kuratowski’s theorem or Wagner’s theorem to show that the graph is nonplanar.

Note: It may help to review our definitions of PG and MPG on the previous page, as we are using these terms a little differently than the standard usage of “planar graph” or “maximal planar graph.”

## 数学代写|图论作业代写Graph Theory代考|Challenge problem

Challenge problem 1: Since $\mathrm{K}{3,3}$ is two-colorable, whereas $\mathrm{K}{5}$ isn’t fourcolorable, it may seem natural to wonder if either Kuratowski’s theorem or Wagner’s theorem may be applied in some way to prove the four-color theorem. For example, one might wonder if every graph that doesn’t contain a subgraph that is a subdivision of $\mathrm{K}{5}$ may be four-colorable, or if every graph whose minors don’t contain $\mathrm{K}{5}$ is four-colorable. Some of these graphs wouldn’t be MPG’s, but that’s not a problem. If every graph that meets the specified criteria can be shown to be four-colorable and if the set of graphs includes all possible MPG’s, it doesn’t matter if the set of graphs also includes some non-planar graphs as well.
Do one of the following:

• Show that such an idea doesn’t work by providing a counterexample (such as a graph that doesn’t contain a subgraph that is a subdivision of $\mathrm{K}_{5}$ which isn’t four-colorable).
• Explain why it would be impossible or very difficult to prove the fourcolor theorem with this approach.
• Formulate a proof of the four-color theorem using this idea. (Before you choose this option, consider that Kuratowski’s theorem has been known for nearly a century, but as of the publication of this book no attempts to prove the four-color theorem by hand have been accepted by the mathematics community; the only accepted proof involves computer calculations.)
Note: The answer key doesn’t include answers to the challenge problems. These problems are intended to encourage you to think about the ideas. However, you may want to consider how this problem relates to the challenge problems from Chapters 10 and 16, and how this problem relates to Chapter $27 .$

## 数学代写|图论作业代写Graph Theory代KEMPE CHAINS

In 1879, Alfred Kempe recognized that pairs of colors in PG’s make strings called Kempe chains [Ref. 2]. For example, the MPG below shows R-Y Kempe chains. Most of the shaded vertices below participate in one very long section of the R-Y Kempe chains. (If a chain appears to end near the top, right, bottom, or left, it may continue along one of the “outside” edges.) There are also a few short sections of R-Y Kempe chains in the figure below:

• Near the bottom center, there is a short chain with just one $R$ and one Y.
• Near the top left, there is a lone R surrounded by blues and greens.
• Toward the right, a couple of rows up is a lone $Y$ surrounded by blues and greens.
• At the bottom, a few columns from the left is another lone Y.
Note that the R-Y at the bottom right is actually part of the main, very long section.

There are also B-G Kempe chains that complement the R-Y Kempe chains. If you focus on the non-shaded vertices, you will see the B-G Kempe chains. There are three lengthy sections of B-G Kempe chains and one short section at the top right in the MPG below. Each section of a Kempe chain is isolated from the other sections of the same color pair. For example, examine the B-G Kempe chains on the first graph of this chapter, which has three lengthy sections and one short section. Note how these four sections of B-G Kempe chains are isolated from one another by the R-Y Kempe chains. (The sections of R-Y Kempe chains are similarly isolated from one another by the B-G Kempe chains. There is a similar relationship between the two color pairs of the other two figures. This relationship is characteristic of all PG’s.)

## 数学代写|图论作业代写Graph Theory代考|EXERCISES

1. 对于下面和下一页的每个图表，首先使用欧拉公式应用排除测试以确定以下之一：
• 该图绝对是非平面的。
• 该图可以是 MPG。
• 该图可以是 PG，但不是 MPG。
现在确定每个图形是 MPG、PG 但不是 MPG 还是非平面的。如果图形是 MPG 或 PG，则重新绘制图形以表明图形是 PG。如果图是非平面的，则应用 Kuratowski 定理或 Wagner 定理来证明该图是非平面的。

## 数学代写|图论作业代写Graph Theory代考|Challenge problem

• 通过提供一个反例来证明这样的想法是行不通的（例如一个不包含一个子图的图，它是ķ5这不是四色的）。
• 解释为什么用这种方法证明四色定理是不可能或非常困难的。
• 用这个想法来证明四色定理。（在您选择此选项之前，请考虑 Kuratowski 定理已为人所知近一个世纪，但截至本书出版时，数学界尚未接受任何手动证明四色定理的尝试；唯一被接受的证明涉及计算机计算。）
注意：答案键不包括挑战问题的答案。这些问题旨在鼓励您思考这些想法。但是，您可能需要考虑这个问题如何与第 10 章和第 16 章中的挑战问题相关，以及该问题如何与第27.

## 数学代写|图论作业代写Graph Theory代KEMPE CHAINS

1879 年，Alfred Kempe 认识到 PG 中的颜色对构成称为 Kempe 链的字符串 [Ref. 2]。例如，下面的 MPG 显示了 RY Kempe 链。下面的大多数阴影顶点都参与了 RY Kempe 链的一个非常长的部分。（如果一条链似乎在顶部、右侧、底部或左侧附近结束，它可能会沿着“外”边缘之一继续。）下图中还有一些 RY Kempe 链的短部分：

• 靠近底部中心，有一条短链，只有一条R和一个 Y。
• 在左上角附近，有一个被蓝色和绿色包围的单独的 R。
• 向右，上几排是孤零零的是被蓝色和绿色包围。
• 在底部，从左边算起的几列是另一个单独的 Y。
请注意，右下角的 RY 实际上是主要的非常长的部分的一部分。

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## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。