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

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## 数学代写|图论代写graph theory代考|On-line Coloring

On-line coloring differs from a general coloring in that the vertices are examined one at a time (hence they are seen in a linear manner, or “on a line”). Often, we are restricted to situations where portions of the graph are visible at different times and so a vertex must be assigned a color without all the information available. The notion of an on-line coloring relies on a specific type of subgraph, called an induced subgraph, which was defined on page 6 and discussed in our section on graph coloring results (see page 289). Recall that we need induced subgraphs for coloring problems since if we only took any subgraph and colored it, we may be missing edges that would indicate two vertices need different colors in the larger graph. On-line coloring algorithms require a vertex to be colored based only upon the induced subgraph containing that vertex and the previously colored vertices.

## 数学代写|图论代写graph theory代考|Proof of Brooks’ Theorem

The proof we present of Brooks’ Theorem (Theorem 6.7, restated below) is based on that of Lovász [63] and most closely resembles those in [46] and [84]. This proof is perhaps one of the more complex one we have encountered in this text. Part of the difficulty comes from the number of different scenarios we will address separately. We will first consider if the graph is not regular, then look at its connectivity. Within each case we will be creating an order of the vertices so that each vertex has at most $\Delta(G)-1$ neighbors preceding it.
Theorem 6.7 (Brooks’ Theorem) Let $G$ be a connected graph and $\Delta$ denote the maximum degree among all vertices in $G$. Then $\chi(G) \leq \Delta$ as long as $G$ is not a complete graph or an odd cycle. If $G$ is a complete graph or an odd cycle then $\chi(G)=\Delta+1$.

## 数学代写|图论代写GRAPH THEORY代考|Weighted Coloring

Consider the following scenario:
Ten families need to buy train tickets for an upcoming trip. The families vary in size but each of them needs to sit together on the train. Determine the minimum number of seats needed to accommodate the ten family trips.
This problem should sound very similar to Example $6.15$ where colors were representing seats and the vertices were intervals of time indicative of when a person was on the train. Here, we are still interested in a graph coloring, but now each vertex represents a family and so has a size associated with it. In previous chapters, we used weights on the edges of a graph to indicate distance, time, or cost. For graph coloring models, weighted edges would have very little meaning. Instead, we will assigning a weight to each vertex, and finding a proper coloring will be referred to as a weighted coloring.

Given a weighted graph $G=(V, E, w)$, where $w$ assigns each vertex a positive integer, a proper weighted coloring of $G$ assigns each vertex a set of colors so that
(i) the set consists of consecutive colors (or numbers);
(ii) the number of colors assigned to a vertex equals its weight; and
(iii) if two vertices are adjacent, then their set of colors must be disjoint.

## 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。