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# 数学代写|图论代写graph theory代考|Edge Coloring

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## 数学代写|图论代写graph theory代考|1-Factorizations Revisited

Recall that in Section $5.4$ we discussed a variation on matching in which the edges of a graph would be partitioned into disjoint 1-factors, called a 1-factorization. If we give each edge of a 1-factor the same color, then a 1-factorization could be represented as an edge-coloring of a graph. However, not every edge-coloring can represent a 1-factorization, since a 1-factor must also span the graph, requiring every vertex to be incident to an edge of every color. Thus we need a graph to be $k$-regular and have chromatic index $k$ to have a 1-factorization. The graph $G_{5}$ below is 4-regular with chromatic index 4 and contains a 1-factorization.

## 数学代写|图论代写graph theory代考|Ramsey Numbers

As with vertex-coloring, when investigating edge-colorings we are often concerned with finding an optimal coloring (using the least number of colors possible). Other problems exist, where optimality is no longer the goal. One such edge-coloring problem relaxes some of the restrictions on coloring the edges, and is named for the British mathematician and economist, Frank P. Ramsey. Ramsey’s legacy is less so due to his own publications, mainly due to his death at the age of 26 , but rather for the many theories and results that arose from his limited publications. In particular a minor lemma in his 1928 paper “On a problem of formal logic” stated that within any system there exists some underlying order [71]. Although a simple concept, it birthed an area of mathematics now known as Ramsey Theory.

Ramsey Theory can be described in different forms, so we will naturally use the graph theoretic version. In particular, we will discuss Ramsey numbers as they relate to coloring the edges of a graph. Unlike our edge-colorings above in which no two edges can be given the same color if they have a common endpoint, here we will be concerned with specific monochromatic structures within the larger graph.

Given positive integers $m$ and $n$, the Ramsey number $R(m, n)$ is the minimum number of vertices $r$ so that all simple graphs on $r$ vertices contain either a clique of size $m$ or an independent set of size $n$.

## 数学代写|图论代写GRAPH THEORY代考|RAMSEY NUMBERS

Ramsey 理论可以用不同的形式来描述，所以我们自然会使用图论版本。特别是，我们将讨论 Ramsey 数，因为它们与为图的边缘着色有关。与我们上面的边着色不同，如果两条边有一个共同的端点，就不能赋予它们相同的颜色，这里我们将关注较大图中的特定单色结构。

## 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。