如果你也在 怎样代写图论Graph Theory 这个学科遇到相关的难题，请随时右上角联系我们的24/7代写客服。图论Graph Theory有趣的部分原因在于，图可以用来对某些问题中的情况进行建模。这些问题可以在图表的帮助下进行研究(并可能得到解决)。因此，图形模型在本书中经常出现。然而，图论是数学的一个领域，因此涉及数学思想的研究-概念和它们之间的联系。我们选择包含的主题和结果是因为我们认为它们有趣、重要和/或代表主题。

图论Graph Theory通过熟悉许多过去和现在对图论的发展负责的人，可以增强对图论的欣赏。因此，我们收录了一些关于“图论人士”的有趣评论。因为我们相信这些人是图论故事的一部分，所以我们在文中讨论了他们，而不仅仅是作为脚注。我们常常没有认识到数学是一门有生命的学科。图论是人类创造的，是一门仍在不断发展的学科。

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数学代写|图论作业代写Graph Theory代考|Perfect graphs

As discussed in Section 5.2, a high chromatic number may occur as a purely global phenomenon: even when a graph has large girth, and thus locally looks like a tree, its chromatic number may be arbitrarily high. Since such ‘global dependence’ is obviously difficult to deal with, one may become interested in graphs where this phenomenon does not occur, i.e. whose chromatic number is high only when there is a local reason for it.
Before we make this precise, let us note two definitions for a graph $G$. The greatest integer $r$ such that $K^r \subseteq G$ is the clique number $\omega(G)$ of $G$, and the greatest integer $r$ such that $\overline{K^r} \subseteq G$ (induced) is the independence number $\alpha(G)$ of $G$. Clearly, $\alpha(G)=\omega(\bar{G})$ and $\omega(G)=\alpha(\bar{G})$.
A graph is called perfect if every induced subgraph $H \subseteq G$ has chromatic number $\chi(H)=\omega(H)$, i.e. if the trivial lower bound of $\omega(H)$ colours always suffices to colour the vertices of $H$. Thus, while proving an assertion of the form $\chi(G)>k$ may in general be difficult, even in principle, for a given graph $G$, it can always be done for a perfect graph simply by exhibiting some $K^{k+1}$ subgraph as a ‘certificate’ for non-colourability with $k$ colours.

At first glance, the structure of the class of perfect graphs appears somewhat contrived: although it is closed under induced subgraphs (if only by explicit definition), it is not closed under taking general subgraphs or supergraphs, let alone minors (examples?). However, perfection is an important notion in graph theory: the fact that several fundamental classes of graphs are perfect (as if by fluke) may serve as a superficial indication of this. ${ }^3$

What graphs, then, are perfect? Bipartite graphs are, for instance. Less trivially, the complements of bipartite graphs are perfect, tooa fact equivalent to König’s duality theorem 2.1.1 (Exercise 36). The so-called comparability graphs are perfect, and so are the interval graphs (see the exercises); both these turn up in numerous applications.

In order to study at least one such example in some detail, we prove here that the chordal graphs are perfect: a graph is chordal (or triangulated) if each of its cycles of length at least 4 has a chord, i.e. if it contains no induced cycles other than triangles.

To show that chordal graphs are perfect, we shall first characterize their structure. If $G$ is a graph with induced subgraphs $G_1, G_2$ and $S$, such that $G=G_1 \cup G_2$ and $S=G_1 \cap G_2$, we say that $G$ arises from $G_1$ and $G_2$ by pasting these graphs together along $S$.

数学代写|图论作业代写Graph Theory代考|Circulations

In the context of flows, we have to be able to speak about the ‘directions’ of an edge. Since, in a multigraph $G=(V, E)$, an edge $e=x y$ is not identified uniquely by the pair $(x, y)$ or $(y, x)$, we define directed edges as triples:
$$
\vec{E}:={(e, x, y) \mid e \in E ; x, y \in V ; e=x y} .
$$
Thus, an edge $e=x y$ with $x \neq y$ has the two directions $(e, x, y)$ and $(e, y, x)$; a loop $e=x x$ has only one direction, the triple $(e, x, x)$. For given $\vec{e}=(e, x, y) \in \vec{E}$, we set $\bar{e}:=(e, y, x)$, and for an arbitrary set $\vec{F} \subseteq \vec{E}$ of edge directions we put
$$
\bar{F}:={\bar{e} \mid \vec{e} \in \vec{F}}
$$
Note that $\vec{E}$ itself is symmetrical: $\bar{E}=\vec{E}$. For $X, Y \subseteq V$ and $\vec{F} \subseteq \vec{E}$, define
$$
\vec{F}(X, Y):={(e, x, y) \in \vec{F} \mid x \in X ; y \in Y ; x \neq y},
$$
abbreviate $\vec{F}({x}, Y)$ to $\vec{F}(x, Y)$ etc., and write
$$
\vec{F}(x):=\vec{F}(x, V)=\vec{F}({x}, \overline{{x}}) .
$$
Here, as below, $\bar{X}$ denotes the complement $V \backslash X$ of a vertex set $X \subseteq V$. Note that any loops at vertices $x \in X \cap Y$ are disregarded in the definitions of $\vec{F}(X, Y)$ and $\vec{F}(x)$.

Let $H$ be an abelian semigroup, ${ }^2$ written additively with zero 0 . Given vertex sets $X, Y \subseteq V$ and a function $f: \vec{E} \rightarrow H$, let
$$
f(X, Y):=\sum_{\vec{e} \in \vec{E}(X, Y)} f(\vec{e})
$$

图论代考

数学代写|图论作业代写Graph Theory代考|Perfect graphs

正如第5.2节所讨论的，高色数可能作为一种纯粹的全局现象出现:即使一个图有很大的周长，因此局部看起来像一棵树，它的色数可能是任意高的。由于这种“全局依赖”显然很难处理，人们可能会对不发生这种现象的图感兴趣，即只有在存在局部原因的情况下，其色数才高。
在我们明确这一点之前，让我们注意图$G$的两个定义。最大的整数$r$使得$K^r \subseteq G$是$G$的团数$\omega(G)$，最大的整数$r$使得$\overline{K^r} \subseteq G$(诱导)是$G$的独立数$\alpha(G)$。显然是$\alpha(G)=\omega(\bar{G})$和$\omega(G)=\alpha(\bar{G})$。
如果每个诱导子图$H \subseteq G$都有色数$\chi(H)=\omega(H)$，即$\omega(H)$颜色的平凡下界总是足以为$H$的顶点上色，则称为完美图。因此，虽然证明形式为$\chi(G)>k$的断言通常是困难的，即使在原则上，对于给定的图$G$，它总是可以通过简单地展示一些$K^{k+1}$子图作为具有$k$颜色的不可着色性的“证书”来完成。

乍一看，完美图类的结构似乎有些做作:虽然它在诱导子图下是封闭的(如果仅通过显式定义)，但它在取一般子图或超图时并不封闭，更不用说子图了(例子?)然而，完美性在图论中是一个重要的概念:图的几个基本类是完美的(似乎是侥幸)，这一事实可能是这一点的表面迹象。 ${ }^3$

那么，什么样的图表是完美的呢?例如，二部图。不那么平凡的是，二部图的补也是完美的，这一事实等价于König的对偶定理2.1.1(练习36)。所谓的可比性图是完美的，区间图也是如此(参见练习);这两种情况在许多应用程序中都会出现。

为了详细地研究至少一个这样的例子，我们在这里证明弦图是完美的:一个图是弦图(或三角图)，如果它的每个长度至少为4的环都有一个弦，即，如果它不包含除三角形以外的诱导环。

为了证明弦图是完美的，我们将首先描述弦图的结构。如果$G$是一个具有诱导子图$G_1, G_2$和$S$的图，例如$G=G_1 \cup G_2$和$S=G_1 \cap G_2$，我们说$G$是由$G_1$和$G_2$通过沿着$S$粘贴在一起而产生的。

数学代写|图论作业代写Graph Theory代考|Circulations

在流动的背景下，我们必须能够谈论边缘的“方向”。由于在多图$G=(V, E)$中，边$e=x y$不是由$(x, y)$或$(y, x)$对唯一标识的，因此我们将有向边定义为三元组:
$$
\vec{E}:={(e, x, y) \mid e \in E ; x, y \in V ; e=x y} .
$$
因此，具有$x \neq y$的边$e=x y$具有$(e, x, y)$和$(e, y, x)$两个方向;一个循环$e=x x$只有一个方向，即三重$(e, x, x)$。对于给定的$\vec{e}=(e, x, y) \in \vec{E}$，我们设置$\bar{e}:=(e, y, x)$，对于任意的边方向集$\vec{F} \subseteq \vec{E}$，我们设置
$$
\bar{F}:={\bar{e} \mid \vec{e} \in \vec{F}}
$$
注意$\vec{E}$本身是对称的:$\bar{E}=\vec{E}$。对于$X, Y \subseteq V$和$\vec{F} \subseteq \vec{E}$，定义
$$
\vec{F}(X, Y):={(e, x, y) \in \vec{F} \mid x \in X ; y \in Y ; x \neq y},
$$
将$\vec{F}({x}, Y)$缩写为$\vec{F}(x, Y)$等，并写上
$$
\vec{F}(x):=\vec{F}(x, V)=\vec{F}({x}, \overline{{x}}) .
$$
下面，$\bar{X}$表示顶点集$X \subseteq V$的补$V \backslash X$。注意，在$\vec{F}(X, Y)$和$\vec{F}(x)$的定义中，顶点$x \in X \cap Y$处的任何循环都将被忽略。

设$H$是一个阿贝尔半群，${ }^2$与0相加。给定顶点集$X, Y \subseteq V$和函数$f: \vec{E} \rightarrow H$，设
$$
f(X, Y):=\sum_{\vec{e} \in \vec{E}(X, Y)} f(\vec{e})
$$

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