# 数学代写|图论作业代写Graph Theory代考|MATH141A

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

When designing electrical circuits, such as the circuit board within a computer, it is important that wires do not cross. As we have seen, if there are too many connections between the points then crossings are necessary on a plane. To combat this, we can break up the wires into different layers, each one of which contains no crossings. Since each new layer would incur additional cost, we want to minimize the number of layers necessary. In graph theoretic terms, we want to decompose the graph into spanning subgraphs, each of which are planar, using the smallest number of subgraphs possible. This minimum value is called the thickness of a graph.

Definition 7.22 Let $T=\left{H_1, H_2, \ldots, H_t\right}$ be a set of spanning subgraphs of $G$ so that each $H_i$ is planar and every edge of $G$ appears in exactly one graph from $T$. The thickness of a graph $G$, denoted $\theta(G)$, is the minimum size of $T$ among all possible such collections.

Clearly $\theta(G)=1$ if and only if $G$ is planar, since $T$ would contain only $G$ itself. Below is a decomposition of $K_6$ into two planar spanning subgraphs and since we know that $K_6$ is not planar we have shown $\theta\left(K_6\right)=2$.

Corollary 7.23 Let $G$ be a connected simple graph with $n$ vertices and $m$ edges. Then
$$\theta(G) \geq\left\lceil\frac{m}{3 n-6}\right\rceil$$
Corollary 7.24 Let $G$ be a connected simple bipartite graph with $n$ vertices and $m$ edges. Then
$$\theta(G) \geq\left\lceil\frac{m}{2 n-4}\right\rceil$$
While a general formula is not known for the thickness of a graph, the theorem below does establish the thickness for a complete graph (and so could serve as an upper bound for any graph on $n$ vertices). This result is based on the work from $[1],[3],[4]$, and $[81]$.

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

While graphs are defined in terms of a set of vertices and edges, and these two sets uniquely determine the graph, this section will focus more generally on the basics of set theory.

Definition A.1 A set is a collection of objects. We denote sets with capital letters $(A, B, C, \ldots)$ and the objects within a set $A$ are often called elements, denoted $x \in A$. If $x$ is not an element of $A$ we write $x \notin A$.
Suppose $A$ is the set of digits, that is the set of integers from 0 to 9 . We could write this as
$$A={0,1,2,3,4,5,6,7,8,9}$$
which is called roster notation since we are explicitly listing the elements in the set. We could also write $A$ in set-builder notation as follows:
$$A={x \in \mathbb{Z}: 0 \leq x \leq 9}$$
Here we are giving a rule for when an element belongs in a set. The advantage of set-builder notation is when there are a large number of elements in the set and listing them would be excessively time consuming; however, this only works when there is a clear way to describe the elements in the set. For small sets, roster notation is preferred.

There are numerous operations one can perform on a set. We will be concerned with a handful of these, mainly unions, intersections, complements, and subsets. To do this, we must first define our domain for a given problem, which is called the universal set.

# 图论代考

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

$$\theta(G) \geq\left\lceil\frac{m}{3 n-6}\right\rceil$$

$$\theta(G) \geq\left\lceil\frac{m}{2 n-4}\right\rceil$$

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

$$A={0,1,2,3,4,5,6,7,8,9}$$

$$A={x \in \mathbb{Z}: 0 \leq x \leq 9}$$

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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。

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