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数学代写|图论代考GRAPH THEORY代写|Four Color Theorem

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数学代写|图论代写GRAPH THEORY代考|Four Color Theorem

In 1852 Augustus De Morgan sent a letter to his colleague Sir William Hamilton (the same mathematician who introduced what we now call hamiltonian cycles) regarding a puzzle presented by one of his students, Frederick Gutherie (though Gutherie later clarified that the question originated from his brother, Francis). This question was known for over a century as the Four Color Conjecture, and can be stated as

Any map split into contiguous regions can be colored using at most four colors so that no two bordering regions are given the same color.

An important aspect of this conjecture is that a region, such as a country or state, cannot be split into two disconnected pieces. For example, the state of Michigan is split into the Lower Peninsula and the Upper Peninsula and so is not a contiguous region; thus the contiguous United States does not satisfy the hypothesis of the Four Color Conjecture. However, it is still possible to color the lower 48 states using 4 colors (try it!).

The Four Color Conjecture started as a map coloring problem, yet migrated into a graph coloring problem. In the late 19th century, Alfred Kempe studied the dual problem where each region on a map was represented by a vertex and an edge exists between two vertices if their corresponding regions share a border. This approach was extensively used in the mid-20th century as the study of graph theory exploded with the advent of the computer. The search for a proper map coloring is now reduced to a proper vertex coloring (more commonly referred to as just a coloring) for a planar graph. A graph is planar if it can be drawn so that no edges cross. We will study planar graphs extensively in Chapter 7.

数学代写|图论代写GRAPH THEORY代考|Vertex Coloring

For the remainder of this chapter, we will explore graph colorings for graphs that may or may not be planar, mainly since we already know that planar graphs need at most 4 colors and so there is not much room for further exploration. Any graph we consider can be simple or have multi-edges but cannot have loops, since a vertex with a loop could never be assigned a color. In any graph coloring problem, we want to determine the smallest value for $k$ for which a graph has a $k$-coloring. This value for $k$ is called the chromatic number of a graph.

Definition 6.4 The chromatic number $\chi(G)$ of a graph is the smallest value $k$ for which $G$ has a proper $k$-coloring.

In order to determine the chromatic number of a graph, we often need to complete the following two steps:
(1) Find a vertex coloring of $G$ using $k$ colors.
(2) Show why fewer colors will not suffice.
At times it can be quite complex to show a graph cannot be colored with fewer colors. There are a few properties of graphs and the existence of certain subgraphs that can immediately provide a basis for these arguments.

Look back at Example 6.1 about coloring the counties in Vermont and the discussion of alternating colors around a central vertex. In doing so, we were using one of the most basic properties in graph coloring: the number of colors needed to color a cycle. Recall that a cycle on $n$ vertices is denoted $C_n$. The examples below show optimal colorings of $C_3, C_4, C_5$, and $C_6$.

Notice that in all the graphs we try to alternate colors around the cycle. When $n$ is even, we can color $C_n$ in two colors since this alternating pattern can be completed around the cycle. When $n$ is odd, we need three colors for $C_n$ since the final vertex visited when traveling around the cycle will be adjacent to a vertex of color 1 and of color 2 . This was demonstrated in the coloring of the five counties surrounding Lamoille County in Example 6.1.

数学代写|图论代写GRAPH THEORY代考|Four Color Theorem

1852 年，奥古斯都·德·摩根 (Augustus De Morgan) 就他的一位学生弗雷德里克·古瑟里 (Frederick Gutherie) 提出的一个谜题致函他的同事威廉·汉密尔顿爵士（同一位数学家，他提出了我们现在所说的哈密顿循环）（尽管古瑟里后来澄清说这个问题源于他的兄弟，弗朗西斯）。这个问题被称为四色猜想已有一个多世纪了，可以表述为

数学代写|图论代写GRAPH THEORY代考|Vertex Coloring

(1) 找到一个顶点撯色为 $G$ 使用 $k$ 颜色。
(2) 说明为什么更少的颜色是不够的。

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。