## 数学代写|组合数学代写Combinatorial mathematics代考|MATH4575

2022年12月22日

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11.5. We already know that $K_n$ is planar for $n<5$ (Figure $11.1$ shows this for $n=4$ ). Let us take a look at $K_5$. It has $p=5$ vertices and $q=10$ edges. Assume that $K_5$ is planar; then, by Exercise 11.2, it satisfies the inequality $q \leq 3(p-2)$, i.e., $10 \leq 9$, which is absurd. Therefore, $K_5$ is not planar.

Finally, every graph $K_n$ for $n \geq 5$ contains $K_5$ as a subgraph. Therefore, $K_n$ is not planar for $n \geq 5$.
11.6. Figure $11.6$ shows that $K_{m, 2}$ is planar for any positive integer $m$. Therefore any subgraph of $K_{m, 2}$ is planar as well, i.e., a complete bipartite graph $K_{m, n}$ is planar if at least one of the positive integers $m$, $n$ is less than three.

Let us take a look at $K_{3,3}$. It has $p=6$ vertices, $q=9$ edges, and no triangles (Theorem 9.2). Therefore, by Exercise $11.3^{\prime}, q \leq$ $2(p-2)$, i.e., $9 \leq 8$, which is absurd. Therefore, $K_{3,3}$ is not planar.
Every graph $K_{m, n}$ for $m \geq 3, n \geq 3$ contains $K_{3,3}$ as a subgraph, and therefore is not planar.
11.7. Both graphs $G_1$ and $G_2$ can be obtained from the graph $G$ (Figure 11.7) by a succession of elementary subdivisions.

## 数学代写|组合数学代写Combinatorial mathematics代考|The Intersection Index and the Jordan Curve Theorem

In the previous section we talked about the regions we obtain when a graph is drawn in the plane without intersections. Is it clear what a region is? Perhaps you would answer yes. But allow us to shake your confidence in the clarity of this notion.

Let us consider a very simple example: look at a complete graph $K_3$, the contour of a triangle (Figure 12.1). Is it evident that the graph in Figure $12.1$ divides the plane into two regions? In other words, is it true that every closed curve that does not intersect itself defines two regions, interior and exterior?

It appears “visually obvious” that a curve $C$ cannot be a boundary common to more than two regions in the plane such that all regions border $C$ along its entirety. Intuition, however, can trick us.

Example 12.1. There is a curve in the plane that is a common boundary of three regions.

Some such curves, known as the “Lakes of Wada,” were discovered by the Japanese mathematician Kunizô Yoneyama in 1917 ([Y]).
Proof. Assume that a portion of land surrounded by the sea contains two lakes: a warm lake and a cold lake. In order to provide the land with water we build canals.

On the first day, we build a canal (Figure 12.2) that delivers warm lake water so that it is available at a distance not exceeding 1 from every point of land (this canal is neither connected to the sea nor to the cold lake!).

On the second day, we build a canal that delivers cold lake water so that it is available at a distance not exceeding 1 from every point of the remaining land (this canal is not connected to the sea, the warm lake, or to the previously built canal).

On the third day, we build a canal that delivers sea water so that it is available at a distance not exceeding 1 from every point of the remaining land (of course, this canal is not connected to the lakes or to the previously built canals).

During the next three days we extend the three canals further so that the warm lake water, the cold lake water, and the sea water are available from any point of the remaining land at a distance not exceeding $\frac{1}{2}$.

## 组合数学代考

11.5。我们已经知道 $K_n$ 在 $n<5$ 时是平面的（图 $11.1$ 显示 $n=4$ 时）。让我们来看看$K_5$。它有 $p=5$ 个顶点和 $q=10$ 个边。假设 $K_5$ 是平面的；然后，通过练习 11.2，它满足不等式 $q \leq 3(p-2)$，即 $10 \leq 9$，这是荒谬的。因此，$K_5$ 不是平面的。

11.6. 图 $11.6$ 显示 $K_{m, 2}$ 对于任何正整数 $m$ 都是平面的。因此 $K_{m, 2}$ 的任何子图也是平面的，即一个完整的二分图 $K_{m, n}$ 是平面的，如果至少一个正整数 $m$, $n$ 小于比三个。

$K_{m, n}$ for $m \geq 3, n \geq 3$ 包含 $K_{3,3}$ 作为子图，因此不是平面的。
11.7. 图 $G_1$ 和 $G_2$ 都可以通过一系列基本细分从图 $G$（图 11.7）中获得。

## 数学代写|组合数学代写Combinatorial mathematics代考|The Intersection Index and the Jordan Curve Theorem

1917 年，日本数学家米山国三 ([Y]) 发现了一些被称为“和田湖”的曲线。

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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。