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

statistics-lab™ 为您的留学生涯保驾护航 在代写图论Graph Theory方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写图论Graph Theory代写方面经验极为丰富，各种代写图论Graph Theory相关的作业也就用不着说。

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

In addition to coloring the regions of a map and coloring the vertices of a graph, it is also of interest to color the edges of a graph. An edge coloring of a nonempty graph $G$ is an assignment of colors to the edges of $G$, one color to each edge, such that adjacent edges are assigned different colors. The minimum number of colors that can be used to color the edges of $G$ is called the chromatic index (or sometimes the edge chromatic number) and is denoted by $\chi^{\prime}(G)$. An edge coloring that uses $k$ colors is a $k$-edge coloring. In Figure 10.15, a 4-edge coloring of a graph $G$ is given.

Let $G$ be a graph containing a vertex $v$ with $\operatorname{deg} v=k \geq 1$. Then there are $k$ edges incident with $v$. Any edge coloring must assign $k$ distinct colors to the edges incident with $v$ and $\operatorname{so} \chi^{\prime}(G) \geq \operatorname{deg} v=k$. In particular,
$$\chi^{\prime}(G) \geq \Delta(G)$$
for every nonempty graph $G$.

## 数学代写|图论作业代写Graph Theory代考|Excursion: The Heawood Map Coloring Theorem

We mentioned that during an 11-year period in the 19th century (1879-1890), the Four Color Theorem was considered to have been verified by Alfred Bray Kempe. However, all this changed in 1890 when Percy John Heawood wrote that he had discovered an error Kempe had made in the way he interchanged colors in what were to be called Kempe chains. It was not accidental that Heawood had read Kempe’s paper. When Arthur Cayley asked, at a meeting of the London Mathematical Society in 1878, for the status of the Four Color Conjecture, Henry Smith was presiding over the meeting. Smith was a Professor of Geometry at Oxford University who would mention this conjecture during his lectures. Soon afterwards, Heawood became a student of Smith and Heawood became interested in this problem after hearing about it from Smith.

In his paper, Heawood produced a counterexample (see Figure 10.22), not to the statement Kempe was trying to prove (the Four Color Theorem) but to the proof Kempe had given. Indeed, Kempe’s proof was quite ingenious and Heawood was able to use Kempe’s technique to show that every map could be colored with five or fewer colors. We’ve seen that this is equivalent to showing that every planar graph can be colored with five or fewer colors.

Proof. Assume, to the contrary, that this statement is false. Then among all planar graphs that are not 5-colorable, let $G$ be the one of smallest order. Since $G$ is not 5-colorable, the order of $G$ is necessarily 6 or more.

# 图论代考

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

$$\chi^{\prime}(G) \geq \Delta(G)$$

﻿

﻿

﻿

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

﻿

## 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。

﻿

The graphs above are incomplete. These figures only show a vertex with degree four (vertex E), its nearest neighbors (A, B, C, and D), and segments of A-C Kempe chains. The entire graphs would also contain several other vertices (especially, more colored the same as B or D) and enough edges to be MPG’s. The left figure has A connected to $C$ in a single section of an A-C Kempe chain (meaning that the vertices of this chain are colored the same as A and C). The left figure shows that this A-C Kempe chain prevents B from connecting to $\mathrm{D}$ with a single section of a B-D Kempe chain. The middle figure has A and C in separate sections of A-C Kempe chains. In this case, B could connect to D with a single section of a B-D Kempe chain. However, since the A and C of the vertex with degree four lie on separate sections, the color of C’s chain can be reversed so that in the vertex with degree four, C is effectively recolored to match A’s color, as shown in the right figure. Similarly, D’s section could be reversed in the left figure so that D is effectively recolored to match B’s color.

Kempe also attempted to demonstrate that vertices with degree five are fourcolorable in his attempt to prove the four-color theorem [Ref. 2], but his argument for vertices with degree five was shown by Heawood in 1890 to be insufficient [Ref. 3]. Let’s explore what happens if we attempt to apply our reasoning for vertices with degree four to a vertex with degree five.

## 数学代写|图论作业代写Graph Theory代考|The previous diagrams

The previous diagrams show that when the two color reversals are performed one at a time in the crossed-chain graph, the first color reversal may break the other chain, allowing the second color reversal to affect the colors of one of F’s neighbors. When we performed the $2-4$ reversal to change B from 2 to 4 , this broke the 1-4 chain. When we then performed the 2-3 reversal to change E from 3, this caused C to change from 3 to 2 . As a result, F remains connected to four different colors; this wasn’t reversed to three as expected.
Unfortunately, you can’t perform both reversals “at the same time” for the following reason. Let’s attempt to perform both reversals “at the same time.” In this crossed-chain diagram, when we swap 2 and 4 on B’s side of the 1-3 chain, one of the 4’s in the 1-4 chain may change into a 2, and when we swap 2 and 3 on E’s side of the 1-4 chain, one of the 3’s in the 1-3 chain may change into a 2 . This is shown in the following figure: one 2 in each chain is shaded gray. Recall that these figures are incomplete; they focus on one vertex (F), its neighbors (A thru E), and Kempe chains. Other vertices and edges are not shown.

Note how one of the 3’s changed into 2 on the left. This can happen when we reverse $\mathrm{C}$ and $\mathrm{E}$ (which were originally 3 and 2 ) on E’s side of the 1-4 chain. Note also how one of the 4’s changed into 2 on the right. This can happen when we reverse B and D (which were originally 2 and 4) outside of the 1-3 chain. Now we see where a problem can occur when attempting to swap the colors of two chains at the same time. If these two 2’s happen to be connected by an edge like the dashed edge shown above, if we perform the double reversal at the same time, this causes two vertices of the same color to share an edge, which isn’t allowed. We’ll revisit Kempe’s strategy for coloring a vertex with degree five in Chapter $25 .$

## 数学代写|图论作业代写Graph Theory代考|The shading of one section of the B-R

• MPG 是三角测量的。它由具有三个边和三个顶点的面组成。
• 每个面的三个顶点必须是三种不同的颜色。
• 每条边由两个相邻的三角形共享，形成一个四边形。
• 每个四边形将有 3 或 4 种不同的颜色。如果与共享边相对的两个顶点恰好是相同的颜色，则它有 3 种颜色。
• 对于每个四边形，四个顶点中的至少 1 个顶点和最多 3 个顶点具有任何颜色对的颜色。例如，具有 R、G、B 和G有 1 个顶点R−是和3个顶点乙−G，或者您可以将其视为 1 个顶点乙−是和3个顶点G−R，或者您可以将其视为 BR 的 2 个顶点和 GY 的 2 个顶点。在后一种情况下，2G’ 不是同一链的连续颜色。
• 当您将更多三角形组合在一起（四边形仅组合两个）并考虑可能的颜色时，您将看到 Kempe 的部分

• 画一张R顶点和一个是由边连接的顶点。
• 如果一个新顶点连接到这些顶点中的每一个，它必须是乙或者G.
• 如果一个新顶点连接到 R 而不是是，可能是是,乙， 或者G.
• 如果一个新的顶点连接到是但不是R，可能是R,乙， 或者G.
• RY 链要么继续增长，要么被 B 包围，G.
• 如果你关注 B 和 G，你会为它的链条得出类似的结论。
• 如果一条链条完全被其对应物包围，则链条的新部分可能会出现在其对应物的另一侧。
Kempe 证明了所有具有四阶的顶点（那些恰好连接到其他四个顶点的顶点）都是四色的 [Ref. 2]。例如，考虑下面的中心顶点。

## 数学代写|图论作业代写Graph Theory代考|In the previous figure

• A 和 C 或者是 AC Kempe 链的同一部分的一部分，或者它们各自位于 AC Kempe 链的不同部分。（如果一种和C例如，是红色和黄色的，则 AC 链是红黄色链。） – 如果一种和C每个位于 AC Kempe 链的不同部分，其中一个部分的颜色可以反转，这有效地重新着色 C 以匹配 A 的颜色。如果 A 和 C 是 AC Kempe 链的同一部分的一部分，则 B 和 D每个都必须位于 BD Kempe 链的不同部分，因为 AC Kempe 链将阻止任何 BD Kempe 链从 B 到达 D。（如果乙和D是蓝色和绿色，例如，那么一种BD Kempe 链是蓝绿色链。）在这种情况下，由于 B 和 D 分别位于 BD Kempe 链的不同部分，因此 BD Kempe 链的其中一个部分的颜色可以反转，这有效地重新着色 D 以匹配 B颜色。– 因此，可以使 C 与 A 具有相同的颜色或使 D 具有与 A 相同的颜色乙通过反转 Kempe 链的分离部分。

Kempe 还试图证明五阶顶点是可四色的，以证明四色定理 [Ref. 2]，但 Heawood 在 1890 年证明他关于五次顶点的论点是不充分的 [Ref. 3]。让我们探讨一下如果我们尝试将我们对度数为四的顶点的推理应用于度数为五的顶点会发生什么。

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

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

statistics-lab™ 为您的留学生涯保驾护航 在代写图论Graph Theory方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写图论Graph Theory代写方面经验极为丰富，各种代写图论Graph Theory相关的作业也就用不着说。

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

The directors of an amusement center have decided to open a new theme park in the center. The initial plan for the theme park is to build six attractions, which are temporarily denoted by A1, A2, .., A6. Figure 9.1(a) shows the initial location of the attractions.

In the summer, the amusement center often becomes very hot and walking between attractions can be uncomfortable. Preliminary studies indicate that the least amount of traffic is likely to occur between attractions (1) A1 and A4, (2) A2 and A5 and (3) A3 and A6. The designers feel that, despite the expense, it would be good for business to build an air-conditioned tube enclosing moving walkways in both directions between all pairs of attractions except those in (1)- (3). One possible concern is whether this can be done without any two tubes interfering with each other. Figure 9.1(b) shows that the tubes can indeed be built without any pair intersecting. Figure 9.1(c) shows that if the attractions are relocated, then an even better design for the location of the tubes can be given.

After time passes, it is decided that the attractions A1, A2, .., A6 need to be modified and they are now called B1, B2, .., B6. Furthermore, it is decided to add a seventh attraction B7. (See Figure 9.2.) In addition, it is decided that moving walkway tubes should be built between every pair of attractions, except the pairs ${\mathrm{B} 1, \mathrm{~B} 4},{\mathrm{B} 1, \mathrm{~B} 5},{\mathrm{B} 2, \mathrm{~B} 5},{\mathrm{B} 2, \mathrm{~B} 6},{\mathrm{B} 3, \mathrm{~B} 6},{\mathrm{B} 3, \mathrm{~B} 7}$ and ${\mathrm{B} 4$, B7}. How should this be done?

## 数学代写|图论作业代写Graph Theory代考|Embedding Graphs on Surfaces

If $G$ is a planar graph, then we know that $G$ can be drawn in the plane in such a way that no two edges
cross. Such a “drawing” is also called an embedding of $G$ in the plane. In addition, we say that $G$ can be embedded in the plane. On the other hand, if $G$ is nonplanar, then $G$ cannot be embedded in the plane, that is, it is impossible to draw $G$ in the plane without some of its edges crossing.

Perhaps it is clear that if a graph $G$ is planar, then $G$ can be embedded on the sphere as well as the plane. Furthermore, if a graph $G$ can be embedded on a sphere, then it must be planar. Although these observations may not seem particularly enlightening, this brings up the question of considering surfaces other than the sphere on which a graph might be embedded. But what other surfaces are there? A common surface is the torus, a doughnut-shaped surface (see Figure 9.19(a)). In Figure 9.19(b), we see that the graph $K_4$ can be embedded on the torus. In fact, there is more than one way to embed $K_4$ on the torus (see Figure 9.19(c)).

Not only $\operatorname{can} K_4$ be embedded on the torus, so can $K_5$. Figure 9.20 (a) shows an embedding of $K_5$ on the torus; Figure 9.20(b) shows an embedding of $K_{3,3}$ on the torus.

Embedding graphs on a torus, as we did in Figure 9.20, can be difficult to visualize. However, there are alternative ways to represent these embeddings as we will now explain. How is a torus constructed? One way is to begin with a rectangular piece of material (the more flexible the better) as in Figure 9.21 and first make a cylinder from it by identifying sides $a$ and $c$, which are the same after the identification occurs. The sides $b$ and $d$ then become circles. These circles are then identified to produce a torus.

# 图论代考

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﻿

﻿

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

﻿

## 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。

﻿

The graphs above are incomplete. These figures only show a vertex with degree four (vertex E), its nearest neighbors (A, B, C, and D), and segments of A-C Kempe chains. The entire graphs would also contain several other vertices (especially, more colored the same as B or D) and enough edges to be MPG’s. The left figure has A connected to $C$ in a single section of an A-C Kempe chain (meaning that the vertices of this chain are colored the same as A and C). The left figure shows that this A-C Kempe chain prevents B from connecting to $\mathrm{D}$ with a single section of a B-D Kempe chain. The middle figure has A and C in separate sections of A-C Kempe chains. In this case, B could connect to D with a single section of a B-D Kempe chain. However, since the A and C of the vertex with degree four lie on separate sections, the color of C’s chain can be reversed so that in the vertex with degree four, C is effectively recolored to match A’s color, as shown in the right figure. Similarly, D’s section could be reversed in the left figure so that D is effectively recolored to match B’s color.

Kempe also attempted to demonstrate that vertices with degree five are fourcolorable in his attempt to prove the four-color theorem [Ref. 2], but his argument for vertices with degree five was shown by Heawood in 1890 to be insufficient [Ref. 3]. Let’s explore what happens if we attempt to apply our reasoning for vertices with degree four to a vertex with degree five.

## 数学代写|图论作业代写Graph Theory代考|The previous diagrams

The previous diagrams show that when the two color reversals are performed one at a time in the crossed-chain graph, the first color reversal may break the other chain, allowing the second color reversal to affect the colors of one of F’s neighbors. When we performed the $2-4$ reversal to change B from 2 to 4 , this broke the 1-4 chain. When we then performed the 2-3 reversal to change E from 3, this caused C to change from 3 to 2 . As a result, F remains connected to four different colors; this wasn’t reversed to three as expected.
Unfortunately, you can’t perform both reversals “at the same time” for the following reason. Let’s attempt to perform both reversals “at the same time.” In this crossed-chain diagram, when we swap 2 and 4 on B’s side of the 1-3 chain, one of the 4’s in the 1-4 chain may change into a 2, and when we swap 2 and 3 on E’s side of the 1-4 chain, one of the 3’s in the 1-3 chain may change into a 2 . This is shown in the following figure: one 2 in each chain is shaded gray. Recall that these figures are incomplete; they focus on one vertex (F), its neighbors (A thru E), and Kempe chains. Other vertices and edges are not shown.

Note how one of the 3’s changed into 2 on the left. This can happen when we reverse $\mathrm{C}$ and $\mathrm{E}$ (which were originally 3 and 2 ) on E’s side of the 1-4 chain. Note also how one of the 4’s changed into 2 on the right. This can happen when we reverse B and D (which were originally 2 and 4) outside of the 1-3 chain. Now we see where a problem can occur when attempting to swap the colors of two chains at the same time. If these two 2’s happen to be connected by an edge like the dashed edge shown above, if we perform the double reversal at the same time, this causes two vertices of the same color to share an edge, which isn’t allowed. We’ll revisit Kempe’s strategy for coloring a vertex with degree five in Chapter $25 .$

## 数学代写|图论作业代写Graph Theory代考|The shading of one section of the B-R

• MPG 是三角测量的。它由具有三个边和三个顶点的面组成。
• 每个面的三个顶点必须是三种不同的颜色。
• 每条边由两个相邻的三角形共享，形成一个四边形。
• 每个四边形将有 3 或 4 种不同的颜色。如果与共享边相对的两个顶点恰好是相同的颜色，则它有 3 种颜色。
• 对于每个四边形，四个顶点中的至少 1 个顶点和最多 3 个顶点具有任何颜色对的颜色。例如，具有 R、G、B 和G有 1 个顶点R−是和3个顶点乙−G，或者您可以将其视为 1 个顶点乙−是和3个顶点G−R，或者您可以将其视为 BR 的 2 个顶点和 GY 的 2 个顶点。在后一种情况下，2G’ 不是同一链的连续颜色。
• 当您将更多三角形组合在一起（四边形仅组合两个）并考虑可能的颜色时，您将看到 Kempe 的部分

• 画一张R顶点和一个是由边连接的顶点。
• 如果一个新顶点连接到这些顶点中的每一个，它必须是乙或者G.
• 如果一个新顶点连接到 R 而不是是，可能是是,乙， 或者G.
• 如果一个新的顶点连接到是但不是R，可能是R,乙， 或者G.
• RY 链要么继续增长，要么被 B 包围，G.
• 如果你关注 B 和 G，你会为它的链条得出类似的结论。
• 如果一条链条完全被其对应物包围，则链条的新部分可能会出现在其对应物的另一侧。
Kempe 证明了所有具有四阶的顶点（那些恰好连接到其他四个顶点的顶点）都是四色的 [Ref. 2]。例如，考虑下面的中心顶点。

## 数学代写|图论作业代写Graph Theory代考|In the previous figure

• A 和 C 或者是 AC Kempe 链的同一部分的一部分，或者它们各自位于 AC Kempe 链的不同部分。（如果一种和C例如，是红色和黄色的，则 AC 链是红黄色链。） – 如果一种和C每个位于 AC Kempe 链的不同部分，其中一个部分的颜色可以反转，这有效地重新着色 C 以匹配 A 的颜色。如果 A 和 C 是 AC Kempe 链的同一部分的一部分，则 B 和 D每个都必须位于 BD Kempe 链的不同部分，因为 AC Kempe 链将阻止任何 BD Kempe 链从 B 到达 D。（如果乙和D是蓝色和绿色，例如，那么一种BD Kempe 链是蓝绿色链。）在这种情况下，由于 B 和 D 分别位于 BD Kempe 链的不同部分，因此 BD Kempe 链的其中一个部分的颜色可以反转，这有效地重新着色 D 以匹配 B颜色。– 因此，可以使 C 与 A 具有相同的颜色或使 D 具有与 A 相同的颜色乙通过反转 Kempe 链的分离部分。

Kempe 还试图证明五阶顶点是可四色的，以证明四色定理 [Ref. 2]，但 Heawood 在 1890 年证明他关于五次顶点的论点是不充分的 [Ref. 3]。让我们探讨一下如果我们尝试将我们对度数为四的顶点的推理应用于度数为五的顶点会发生什么。

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

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

statistics-lab™ 为您的留学生涯保驾护航 在代写图论Graph Theory方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写图论Graph Theory代写方面经验极为丰富，各种代写图论Graph Theory相关的作业也就用不着说。

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

A mathematics department at a university has acquired a collection of 12 different mathematics books on a variety of subjects to be presented to students who have performed well on a competitive mathematics exam (one book to each successful student). Of course, there would be a problem if more than 12 students qualified for these books. It turns out, however, that this is not a problem as only 10 students did well enough on the exam to receive books. Nevertheless, another possible difficulty has arisen. Some of the students already have copies of some books and there are some books that certain students have no need for. The question is this: Is there a way of distributing 10 of the 12 books to the 10 students so that each student receives a book that he or she would like to have? The answer to this problem may be no even though there are more books than students. For example, there may be three or more books that no student wants. Also, perhaps there are four students only interested in the same three books, in which case it would be impossible to distribute four books to these four students.

It may already be clear that this situation can be modeled by a graph $G$ whose vertices are the students, say $S_1, S_2, \ldots, S_{10}$ and the books, say $B_1, B_2, \ldots, B_{12}$, where two vertices of $G$ are adjacent if one of these vertices is a student and the other is a book that this student would like to have. Certainly then, $G$ is a bipartite graph with partite sets $U=\left{S_1, S_2, \ldots, S_{10}\right}$ and $W=\left{B_1, B_2, \ldots, B_{12}\right}$. For example, if student $S_1$ would like to have any of the books $B_2, B_3, B_5, B_7$, then the graph $G$ contains the subgraph shown in Figure 8.1. What we are seeking then is a set $A$ of 10 edges in the graph $G$ (where $G$ is only partially drawn in Figure 8.1), no two of which are adjacent. If such a set $A$ exists, then each vertex $S_i(1 \leq i \leq 10)$ is incident with exactly one edge in $A$.

There is a related mathematical question here. Let $U$ and $W$ be two sets such that $|U|=10$ and $|W|=$ 12. Does there exist a one-to-one function $f: U \rightarrow W$ ?

If this is all there is to the question, then the answer is yes. However, what if the image of each element of $U$ cannot be just any element of $W$ ? The image of each element of $U$ is required to be an element of some prescribed subset of $W$. Consequently, what we are asking is that if we know the sets
of possible images of the elements of $U$, is there a one-to-one function $f: U \rightarrow W$ that satisfies these conditions?

This discussion leads us to some new concepts. A set of edges in a graph is independent if no two edges in the set are adjacent. By a matching in a graph $G$, we mean an independent set of edges in $G$. Thus the problem we were discussing asks whether a particular graph contains a certain matching. Since many problems of this type involve bipartite graphs, as does the problem we were discussing, we first consider these concepts for bipartite graphs only.

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

We have mentioned that a matching $M$ in a graph $G$ of order $n$ is a perfect matching if $n$ is even and $|M|=n / 2$. The subgraph $F=G[M]$ induced by $M$ is therefore a 1-regular spanning subgraph of $G$. A 1-regular spanning subgraph of a graph $G$ is also called a 1-factor of $G$. Consequently, the edge set of a 1-factor of a graph is a perfect matching of the graph. So a graph $G$ has a 1-factor if and only if $G$ has a perfect matching.

For even integers $n \geq 4$, the graphs $C_n$ and $K_n$ have 1-factors, while for positive integers $r$ and $s$, the complete bipartite graph $K_{r, s}$ has a 1-factor if and only if $r=s$. The Petersen graph PG (see Figure 8.7) also has a 1-factor, for example, $F=P G[X]$, where $X=\left{u_i u_i: 1 \leq i \leq 5\right}$ is a 1-factor of the Petersen graph. Of course, the Petersen graph is a 3-regular graph. Many other 3-regular graphs have 1-factors. Indeed all of the graphs in Figure 8.7 have 1-factors.

Not every 3-regular graph contains a 1-factor, however. For example, the 3-regular graph $H$ of order 16 shown in Figure 8.8 does not contain a 1-factor. This brings up a question: Which graphs contain 1-factors? Certainly, only graphs of even order can contain a 1-factor. If $G$ is a Hamiltonian graph of even order, then $G$ contains a 1-factor. By taking every other edge in a Hamiltonian cycle, a 1 -factor is obtained. Indeed, a Hamiltonian graph of even order contains two disjoint perfect matchings.

If $G$ is a Hamiltonian graph of even order, then $k(G-S) \leq|S|$ for every nonempty proper subset $S$ of $V(G)$, where, recall, $k(G-S)$ denotes the number of components of $G-S$. This is a consequence of Theorem 6.5. We have seen that the converse of this theorem is not true. For example, $k(P G-S) \leq$ $|S|$ for every nonempty proper subset $S$ of the vertex set of the Petersen graph $P G$ but the Petersen graph is not Hamiltonian. Yet the Petersen graph does contain a 1-factor.

We have already noted that the graph $H$ of Figure 8.8 does not contain a 1 -factor. If it did contain a 1 -factor $F$, then exactly one edge of $F$ is incident with the vertex $v$. Since $H-v$ consists of three components of odd order, two of these components must contain a 1-factor, which, of course, is impossible. This implies that if $G$ is a graph of even order containing a nonempty proper subset $S$ of $V(G)$ such that $G-S$ has more than $|S|$ components of odd order, then $G$ cannot contain a 1-factor. It turns out that this observation is a critical one. A component of a graph is odd or even according to whether its order is odd or even. We write $k_o(G)$ for the number of odd components of a graph $G$. In particular, if $G$ is a Hamiltonian graph of even order $n$ (and thus $G$ contains a 1-factor), then $k_o(G-$ $S) \leq|S|$ for every proper subset $S$ of $V(G)$. The following theorem provides a characterization of graphs containing a 1 -factor.

# 图论代考

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

$$\text { slope }=\frac{\text { rise }}{\text { run }}=\frac{y_2-y_1}{x_2-x_1}$$

$$\text { Slope }=\frac{\text { rise }}{\text { run }}=\frac{10}{5}=2$$

$$\text { Slope }=\frac{y_2-y_1}{x_2-x_1}$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。