### 数学代写|图论作业代写Graph Theory代考|Maps vs. Graphs

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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|Maps vs. Graphs

In a geography class, a map shows relationships between various regions, such as:

• the border surrounding each region
• the size and shape of each region
• which regions share borders with one another
• where one region is located relative to other regions
Notice that we’re using the term region. The regions could be countries on a continent, but they could be states or provinces of a nation or they could be counties that make up a state. The term region allows for generic use. Of the features mentioned above, the only one that we will be concerned with in this book is “which regions share borders with one another.”

In mathematics, especially as it relates to the four-color theorem (which we’ll introduce in Chapter 2), maps and regions have slightly different meanings than they do in geography. For one, we won’t allow a region to consist of two disjointed areas. For example, the United States wouldn’t meet our definition of a region because Alaska and Hawaii are separated from the other 48 states. Another difference between math and geography is that we will require the regions of a map to be contiguous; there can’t be gaps between the regions like lakes. A map doesn’t need to show real places; we will imagine different ways maps can be drawn.For a map, we will use the term edge to refer to any line or curve that separates one region from another region or any line or curve that separates an exterior region from the region outside of the map. Each edge begins at one vertex and ends at another vertex. Every line or curve on a map must adhere to this definition of an edge.For a map, we will use the term vertex to refer to a point where three or more edges intersect. In plural form these points are called vertices. This definition is for a map. (When we learn about graphs, we will see that a vertex has a different definition for a graph, although it will still be a point where edges intersect.)We will require each region of a map as well as the border that surrounds all of the regions to be a simple closed figure. A closed figure divides the plane into two distinct areas: the area inside the figure and the area outside the figure. In contrast, an open figure does not. By simple, we mean that the border of a single region doesn’t cross itself like a figure eight; however, two different regions may join to form a figure eight. Some common examples of simple closed figures include circles, polygons, and ellipses, but the regions don’t need to be common shapes; they just need to be simple closed figures.

## 数学代写|图论作业代写Graph Theory代考|THE FOUR-COLOR

According to the four-color theorem, the vertices of any PG may be colored using no more than four different colors (such as red, blue, green, and yellow) such that [Ref. 1]:

• No two vertices connected by an edge have the same color. (An edge connects two vertices and may be straight, curved, or bent.)
• Every vertex is colored. A single vertex can only be colored using a single color; a multi-colored vertex isn’t allowed.
• The number of vertices is finite. The graph isn’t an infinitely repeated design like a tessellation or fractal. (This particular assumption may not be necessary, but provides a simple starting point with which to approach the four-color theorem.)
• The graph is drawn in the plane or on the surface of a sphere (but other surfaces like a torus are not allowed). Chapter 14 illustrates the concept of the sphere.
• The graph is undirected (the edges don’t have arrows). The graph isn’t disconnected. No edge connects a region to itself (this is called a loop). There are no double edges.
Recall from Chapter 1 that PG stands for “planar graph.” A PG is a graph that can be drawn in the plane without any crossings. $P G$ ‘s are special because they can be mapped in the plane.

Note that there are additional requirements for maps. For example, the regions of a map must be contiguous (there can’t be any gaps or lakes between regions). Two regions of a map may be the same color if they meet only at a vertex (and not an edge). For a map, we can’t allow regions to be disjointed (like the United States, for which Alaska and Hawaii are separated from the other 48 states).

As discussed in Chapter 1 , a single graph may correspond to a multitude of

different maps, which makes it simpler to analyze a graph. For this reason, this book will focus primarily on graphs from this point forward.

Coloring a graph is no different from coloring a map. Below, we colored both a map (lower figures) and its corresponding graph (upper figures). The numbers 1-4 represent four different colors (such as red, blue, green, and yellow). On the graph, no two regions connected by an edge have the same color. On the map, no two regions that share a border have the same color.

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

A graph is triangulated if every face is surrounded by three edges (which may be lines or curves), including the “face” that represents the infinite area “outside” of the graph.

• The left graph below isn’t triangulated because ACDE is a quadrilateral (four-sided).
• The right graph below isn’t triangulated because the infinite area “outside” of the graph has five sides (B, C, D, E, and F) instead of three.
• The center graph below is triangulated because every face, including the infinite area outside, has three sides. Its faces are ABC, ACE, AFE, ABF, BCD, CDE, DEF and BDF. Note that BDF is the infinite area outside (but recall from the end of Chapter 1 that any graph can be inverted to make any face correspond to the infinite area outside of the graph). Any graph that isn’t already triangulated can become triangulated by adding one or more edges to the existing graph. Consider the example below.

In the previous diagram, the left graph isn’t triangulated because ABCF and CDEF each have four sides and because the infinite area outside BGDH also has four sides. If we add edges $\mathrm{AC}, \mathrm{CE}$, and $\mathrm{BD}$ (this one is curved), every face will be a triangle (including the infinite area outside, which is now BDG).

We will use the term maximal planar graph for any PG that has been triangulated in this sense (including the infinite area outside), and we will abbreviate this MPG. An alternative name that is also common is “triangulated graph.” Since every face of a MPG is triangular (in a loose sense of the word, since any of its three edges may be curved), it may seem like triangulated graph would be the better choice. However, since the term triangulated graph is sometimes used with other meanings in mind, the term MPG is common in order to help avoid possible confusion. (We are abbreviating MPG since we will use this term frequently.)

The following property makes it very useful to triangulate graphs to turn PG’s into MPG’s. If a graph is colored in such a way that it satisfies the fourcolor theorem, the same coloring will still satisfy the four-color theorem if one or more edges are removed from the graph. You can see that in the example above. We first colored the MPG on the right. (It turns out that this MPG can be colored using just three colors, but that is unimportant.) We obtained the graph on the left (which is a PG, not a MPG) by removing three edges from the MPG. You can see that the coloring from the MPG on the right still works for the PG on the left after removing the edges from the graph.

Recall from Chapter 1 that a PG is a graph that can be drawn in the plane without crossings. In contrast, a MPG is a special type of PG in that it is fully triangulated, including the infinite area outside.

Any MPG that is colored in such a way as to satisfy the four-color theorem will still satisfy the four-color theorem if any of its edges are removed from the graph. This important classic property of triangulation is the reason that most attempts to prove the four-color theorem only consider MPG’s. If you can prove that the four-color theorem holds are all MPG’s, you will have proven that it holds for all PG’s.

## 数学代写|图论作业代写Graph Theory代考|Maps vs. Graphs

• 每个区域的边界
• 每个区域的大小和形状
• 哪些地区彼此接壤
• 其中一个区域相对于其他区域的位置
请注意，我们正在使用术语区域。这些地区可以是一个大陆上的国家，但它们可以是一个国家的州或省，也可以是组成一个州的县。术语区域允许通用使用。在上面提到的特征中，我们将在本书中关注的唯一一个是“哪些区域彼此共享边界”。

## 数学代写|图论作业代写Graph Theory代考|THE FOUR-COLOR

• 没有两个由边连接的顶点具有相同的颜色。（一条边连接两个顶点，可以是直的、弯曲的或弯曲的。）
• 每个顶点都是彩色的。单个顶点只能使用单一颜色着色；不允许使用多色顶点。
• 顶点的数量是有限的。该图不是像镶嵌或分形那样无限重复的设计。（这个特定的假设可能不是必需的，但提供了一个简单的起点来接近四色定理。）
• 图形绘制在平面内或球体表面（但不允许其他表面，如圆环）。第 14 章说明了球体的概念。
• 该图是无向的（边没有箭头）。图表未断开连接。没有边将区域连接到自身（这称为循环）。没有双刃剑。
回想一下第 1 章，PG 代表“平面图”。PG 是可以在平面上绘制而没有任何交叉的图形。磷G是特殊的，因为它们可以在平面上映射。

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

• 下面的左图没有三角剖分，因为 ACDE 是四边形（四边形）。
• 下面的右图没有三角剖分，因为图“外部”的无限区域有五个边（B、C、D、E 和 F）而不是三个。
• 下面的中心图是三角形的，因为每个面，包括外面的无限区域，都有三个边。它的面是ABC、ACE、AFE、ABF、BCD、CDE、DEF和BDF。请注意，BDF 是外部的无限区域（但回想一下第 1 章末尾的任何图形都可以反转以使任何面对应于图形外部的无限区域）。任何尚未三角化的图都可以通过向现有图添加一条或多条边来进行三角化。考虑下面的例子。

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## MATLAB代写

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