数学代写|图论作业代写Graph Theory代考|MTH607

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数学代写|图论作业代写Graph Theory代考|Ramsey properties and connectivity

According to Ramsey’s theorem, every large enough graph $G$ has a very dense or a very sparse induced subgraph of given order, a $K^r$ or $\overline{K^r}$. If we assume that $G$ is connected, we can say a little more:

Proposition 9.4.1. For every $r \in \mathbb{N}$ there is an $n \in \mathbb{N}$ such that every connected graph of order at least $n$ contains $K^r, K_{1, r}$ or $P^r$ as an induced subgraph.

Proof. Let $d+1$ be the Ramsey number of $r$, let $n:=\frac{d}{d-2}(d-1)^r$, and let $G$ be a graph of order at least $n$. If $G$ has a vertex $v$ of degree at least $d+1$ then, by Theorem 9.1.1 and the choice of $d$, either $N(v)$ induces a $K^r$ in $G$ or ${v} \cup N(v)$ induces a $K_{1, r}$. On the other hand, if $\Delta(G) \leqslant d$, then by Proposition 1.3.3 $G$ has radius $>r$, and hence contains two vertices at a distance $\geqslant r$. Any shortest path in $G$ between these two vertices contains a $P^r$.

In principle, we could now look for a similar set of ‘unavoidable’ $k$-connected subgraphs for any given connectivity $k$. To keep thse ‘unavoidable sets’ small, it helps to relax the containment relation from ‘induced subgraph’ for $k=1$ (as above) to ‘topological minor’ for $k=2$, and on to ‘minor’ for $k=3$ and $k=4$. For larger $k$, no similar results are known.

Proposition 9.4.2. For every $r \in \mathbb{N}$ there is an $n \in \mathbb{N}$ such that every 2-connected graph of order at least $n$ contains $C^r$ or $K_{2, r}$ as a topological minor.

Proof. Let $d$ be the $n$ associated with $r$ in Proposition 9.4.1, and let $G$ be a 2-connected graph with at least $\frac{d}{d-2}(d-1)^r$ vertices. By Proposition 1.3.3, either $G$ has a vertex of degree $>d$ or $\operatorname{diam} G \geqslant \operatorname{rad} G>r$.

In the latter case let $a, b \in G$ be two vertices at distance $>r$. By Menger’s theorem (3.3.6), $G$ contains two independent $a-b$ paths. These form a cycle of length $>r$.

数学代写|图论作业代写Graph Theory代考|Simple sufficient conditions

What kind of condition might be sufficient for the existence of a Hamilton cycle in a graph $G$ ? Purely global assumptions, like high edge density, will not be enough: we cannot do without the local property that every vertex has at least two neighbours. But neither is any large (but constant) minimum degree sufficient: it is easy to find graphs without a Hamilton cycle whose minimum degree exceeds any given constant bound.
The following classic result derives its significance from this background:

Theorem 10.1.1. (Dirac 1952)
Every graph with $n \geqslant 3$ vertices and minimum degree at least $n / 2$ has a Hamilton cycle.

Proof. Let $G=(V, E)$ be a graph with $|G|=n \geqslant 3$ and $\delta(G) \geqslant n / 2$. Then $G$ is connected: otherwise, the degree of any vertex in the smallest component $C$ of $G$ would be less than $|C| \leqslant n / 2$.

Let $P=x_0 \ldots x_k$ be a longest path in $G$. By the maximality of $P$, all the neighbours of $x_0$ and all the neighbours of $x_k$ lie on $P$. Hence at least $n / 2$ of the vertices $x_0, \ldots, x_{k-1}$ are adjacent to $x_k$, and at least $n / 2$ of these same $k<n$ vertices $x_i$ are such that $x_0 x_{i+1} \in E$. By the pigeon hole principle, there is a vertex $x_i$ that has both properties, so we have $x_0 x_{i+1} \in E$ and $x_i x_k \in E$ for some $i<k$ (Fig. 10.1.1).

We claim that the cycle $C:=x_0 x_{i+1} P x_k x_i P x_0$ is a Hamilton cycle of $G$. Indeed, since $G$ is connected, $C$ would otherwise have a neighbour in $G-C$, which could be combined with a spanning path of $C$ into a path longer than $P$.

图论代考

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|Math780

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数学代写|图论作业代写Graph Theory代考|Ramsey numbers

Ramsey’s theorem may be rephrased as follows: if $H=K^r$ and $G$ is a graph with sufficiently many vertices, then either $G$ itself or its complement $\bar{G}$ contains a copy of $H$ as a subgraph. Clearly, the same is true for any graph $H$, simply because $H \subseteq K^h$ for $h:=|H|$.

However, if we ask for the least $n$ such that every graph $G$ of order $n$ has the above property – this is the Ramsey number $R(H)$ of $H$-then the above question makes sense: if $H$ has only few edges, it should embed more easily in $G$ or $\bar{G}$, and we would expect $R(H)$ to be smaller than the Ramsey number $R(h)=R\left(K^h\right)$.

A little more generally, let $R\left(H_1, H_2\right)$ denote the least $n \in \mathbb{N}$ such that $H_1 \subseteq G$ or $H_2 \subseteq \bar{G}$ for every graph $G$ of order $n$. For most graphs $H_1, H_2$, only very rough estimates are known for $R\left(H_1, H_2\right)$. Interestingly, lower bounds given by random graphs (as in Theorem 11.1.3) are often sharper than even the best bounds provided by explicit constructions.

The following proposition describes one of the few cases where exact Ramsey numbers are known for a relatively large class of graphs:

Proposition 9.2.1. Let $s, t$ be positive integers, and let $T$ be a tree of order $t$. Then $R\left(T, K^s\right)=(s-1)(t-1)+1$.

Proof. The disjoint union of $s-1$ graphs $K^{t-1}$ contains no copy of $T$, while the complement of this graph, the complete $(s-1)$-partite graph $K_{t-1}^{s-1}$, does not contain $K^s$. This proves $R\left(T, K^s\right) \geqslant(s-1)(t-1)+1$.
Conversely, let $G$ be any graph of order $n=(s-1)(t-1)+1$ whose complement contains no $K^s$. Then $s>1$, and in any vertex colouring of $G$ (in the sense of Chapter 5) at most $s-1$ vertices can have the same colour. Hence, $\chi(G) \geqslant\lceil n /(s-1)\rceil=t$. By Corollary $5.2 .3, G$ has a subgraph $H$ with $\delta(H) \geqslant t-1$, which by Corollary 1.5 .4 contains a copy of $T$.

数学代写|图论作业代写Graph Theory代考|Induced Ramsey theorems

Ramsey’s theorem can be rephrased as follows. For every graph $H=K^r$ there exists a graph $G$ such that every 2-colouring of the edges of $G$ yields a monochromatic $H \subseteq G$; as it turns out, this is witnessed by any large enough complete graph as $G$. Let us now change the problem slightly and ask for a graph $G$ in which every 2-edge-colouring yields a monochromatic induced $H \subseteq G$, where $H$ is now an arbitrary given graph.

This slight modification changes the character of the problem dramatically. What is needed now is no longer a simple proof that $G$ is ‘big enough’ (as for Theorem 9.1.1), but a careful construction: the construction of a graph that, however we bipartition its edges, contains an induced copy of $H$ with all edges in one partition class. We shall call such a graph a Ramsey graph for $H$.

The fact that such a Ramsey graph exists for every choice of $H$ is one of the fundamental results of graph Ramsey theory. It was proved around 1973 , independently by Deuber, by Erdős, Hajnal \& Pósa, and by Rödl.

Theorem 9.3.1. Every graph has a Ramsey graph. In other words, for every graph $H$ there exists a graph $G$ that, for every partition $\left{E_1, E_2\right}$ of $E(G)$, has an induced subgraph $H$ with $E(H) \subseteq E_1$ or $E(H) \subseteq E_2$.

We give two proofs. Each of these is highly individual, yet each offers a glimpse of true Ramsey theory: the graphs involved are used as hardly more than bricks in the construction, but the edifice is impressive.

图论代考

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|MATH7331

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|The topological end space

In this last section we shall develop a deeper understanding of the global structure of infinite graphs, especially locally finite ones, that can be attained only by studying their ends. This structure is intrinsically topological, but no more than the most basic concepts of point-set topology will be needed.

Our starting point will be to make precise the intuitive idea that the ends of a graph are the ‘points at infinity’ to which its rays converge. To do so, we shall define a topological space $|G|$ associated with a graph $G=(V, E, \Omega)$ and its ends. ${ }^8$ By considering topological versions of paths, cycles and spanning trees in this space, we shall then be able to extend to infinite graphs some parts of finite graph theory that would not otherwise have infinite counterparts (see the notes for more examples). Thus, the ends of an infinite graph turn out to be more than a curious new phenomenon: they form an integral part of the picture, without which it cannot be properly understood.

To build the space $|G|$ formally, we start with the set $V \cup \Omega$. For every edge $e=u v$ we add a set $\dot{e}=(u, v)$ of continuum many points, making these sets $\ddot{e}$ disjoint from each other and from $V \cup \Omega$. We then choose for each $e$ some fixed bijection between $\dot{e}$ and the real interval $(0,1)$, and extend this bijection to one between $[u, v]:={u} \cup \grave{e} \cup{v}$ and $[0,1]$. This bijection defines a metric on $[u, v]$; we call $[u, v]$ a topological edge with inner points $x \in \dot{e}$. Given any $F \subseteq E$ we write $\stackrel{\circ}{F}:=\bigcup{\dot{e} \mid e \in F}$.

When we speak of a ‘graph’ $H \subseteq G$, we shall often also mean its corresponding point set $V(H) \cup \tilde{E}(H)$.

Having thus defined the point set of $|G|$, let us choose a basis of open sets to define its topology. For every edge $u v$, declare as open all subsets of $(u, v)$ that correspond, by our fixed bijection between $(u, v)$ and $(0,1)$, to an open set in $(0,1)$. For every vertex $u$ and $\epsilon>0$, declare as open the ‘open star around $u$ of radius $\epsilon$ ‘, that is, the set of all points on edges $[u, v]$ at distance less than $\epsilon$ from $u$, measured individually for each edge in its metric inherited from $[0,1]$. Finally, for every end $\omega$ and every finite set $S \subseteq V$, there is a unique component $C(S, \omega)$ of $G-S$ that contains a ray from $\omega$. Let $\Omega(S, \omega):=\left{\omega^{\prime} \in \Omega \mid C\left(S, \omega^{\prime}\right)=C(S, \omega)\right}$. For every $\epsilon>0$, write $E_\epsilon(S, \omega)$ for the set of all inner points of $S$ $C(S, \omega)$ edges at distance less than $\epsilon$ from their endpoint in $C(S, \omega)$. Then declare as open all sets of the form
$$\hat{C}\epsilon(S, \omega):=C(S, \omega) \cup \Omega(S, \omega) \cup \dot{E}\epsilon(S, \omega) .$$

数学代写|图论作业代写Graph Theory代考|Ramsey’s original theorems

In its simplest version, Ramsey’s theorem says that, given an integer $r \geqslant 0$, every large enough graph $G$ contains either $K^r$ or $\overline{K^r}$ as an induced subgraph. At first glance, this may seem surprising: after all, we need about $(r-2) /(r-1)$ of all possible edges to force a $K^r$ subgraph in $G$ (Corollary 7.1 .3 ), but neither $G$ nor $\bar{G}$ can be expected to have more than half of all possible edges. However, as the Turán graphs illustrate well, squeezing many edges into $G$ without creating a $K^r$ imposes additional structure on $G$, which may help us find an induced $\overline{K^r}$.

So how could we go about proving Ramsey’s theorem? Let us try to build a $K^r$ or $\overline{K^r}$ in $G$ inductively, starting with an arbitrary vertex $v_1 \in V_1:=V(G)$. If $|G|$ is large, there will be a large set $V_2 \subseteq V_1 \backslash\left{v_1\right}$ of vertices that are either all adjacent to $v_1$ or all non-adjacent to $v_1$. Accordingly, we may think of $v_1$ as the first vertex of a $K^r$ or $\overline{K^r}$ whose other vertices all lie in $V_2$. Let us then choose another vertex $v_2 \in V_2$ for our $K^r$ or $\overline{K^r}$. Since $V_2$ is large, it will have a subset $V_3$, still fairly large, of vertices that are all ‘of the same type’ with respect to $v_2$ as well: either all adjacent or all non-adjacent to it. We then continue our search for vertices inside $V_3$, and so on (Fig. 9.1.1).

How long can we go on in this way? This depends on the size of our initial set $V_1$ : each set $V_i$ has at least half the size of its predecessor $V_{i-1}$, so we shall be able to complete $s$ construction steps if $G$ has order about $2^s$. As the following proof shows, the choice of $s=2 r-3$ vertices $v_i$ suffices to find among them the vertices of a $K^r$ or $\overline{K^r}$.
Theorem 9.1.1. (Ramsey 1930)
For every $r \in \mathbb{N}$ there exists an $n \in \mathbb{N}$ such that every graph of order at least $n$ contains either $K^r$ or $\overline{K^r}$ as an induced subgraph.

Proof. The assertion is trivial for $r \leqslant 1$; we assume that $r \geqslant 2$. Let $n:=2^{2 r-3}$, and let $G$ be a graph of order at least $n$. We shall define a sequence $V_1, \ldots, V_{2 r-2}$ of sets and choose vertices $v_i \in V_i$ with the following properties:
(i) $\left|V_i\right|=2^{2 r-2-i} \quad(i=1, \ldots, 2 r-2)$;

(ii) $V_i \subseteq V_{i-1} \backslash\left{v_{i-1}\right} \quad(i=2, \ldots, 2 r-2)$;
(iii) $v_{i-1}$ is adjacent either to all vertices in $V_i$ or to no vertex in $V_i$ $(i=2, \ldots, 2 r-2)$.

图论代考

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|MATH361

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|Euler’s Formula

There is a simple formula relating the numbers of vertices, edges, and faces in a connected plane graph. It is known as Euler’s Formula because Euler established it for those plane graphs defined by the vertices and edges of polyhedra. In this section, we discuss Euler’s Formula and its immediate consequences.

Theorem 6.3.2 (Euler 1750) Let $G$ be a connected plane graph, and let n, m, and $f$ denote, respectively, the numbers of vertices, edges, and faces of $G$. Then $n-m+f=2$.

Proof We employ an induction on $m$, the result being obvious for $m=0$ or 1 . Assume that $m \geq 2$ and the result is true for all connected plane graphs having fewer than $m$ edges, and suppose that $G$ has $m$ edges. Consider first the case $G$ is a tree. Then $G$ has a vertex $v$ of degree one. The connected plane graph $G-v$ has $n-1$ vertices, $m-1$ edges and $f(=1)$ faces, so by the inductive hypothesis, $(n-1)-(m-1)+f=2$, which implies that $n-m+f=2$. Consider next the case when $G$ is not a tree. Then $G$ has an edge $e$ on a cycle. In this case, the connected plane graph $G-e$ has $n$ vertices, $m-1$ edges, and $f-1$ faces, so that the desired formula immediately follows from the inductive hypothesis.

A maximal planar graph is one to which no edge can be added without losing planarity. Thus in any embedding of a maximal planar graph $G$ with $n \geq 3$, the boundary of every face of $G$ is a triangle, and hence the embedding is often called a triangulated plane graph. Although a general graph may have up to $n(n-1) / 2$ edges, it is not true for planar graphs.

数学代写|图论作业代写Graph Theory代考|Dual Graph

For a plane graph $G$, we often construct another graph $G^$ called the (geometric) dual of $G$ as follows. A vertex $v_i^$ is placed in each face $F_i$ of $G$; these are the vertices of $G^$. Corresponding to each edge $e$ of $G$, we draw an edge $e^$ which crosses $e$ (but no other edge of $G$ ) and joins the vertices $v_i^$ which lie in the faces $F_i$ adjoining $e$; these are the edges of $G^$. The edge $e^$ of $G^$ is called the dual edge of $e$ of $G$. The construction is illustrated in Fig. 6.9; the vertices $v_i^$ are represented by small white circles, and the edges $e^$ of $G^$ by dotted lines. $G^$ is not necessarily a simple graph even if $G$ is simple. Clearly, the dual $G^*$ of a plane graph $G$ is also a plane graph. One can easily observe the following lemma.

Lemma 6.3.6 Let $G$ be a connected plane graph with $n$ vertices, $m$ edges, and $f$ faces, and let the dual $G^$ have $n^$ vertices, $m^$ edges, and $f^$ faces, then $n^=f$, $m^=m$, and $f^*=n$.

Clearly, the dual of the dual of a connected plane graph $G$ is the original graph $G$. However, a planar graph may give rise to two or more geometric duals since the plane embedding is not necessarily unique.

A connected plane graph $G$ is called self-dual if it is isomorphic to its dual $G^$. The graph $G$ in Fig. 6.10 drawn with black vertices and solid edges is a self-dual graph where $G^$ is drawn with white vertices and dotted edges.

A weak dual of a plane graph $G$ is the subgraph of the dual graph of $G$ whose vertices correspond to the inner faces of $G$.

图论代考

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|MATH1230

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数学代写|图论作业代写Graph Theory代考|Dominating Set

For a graph $G=(V, E)$, a set $D \subseteq V(G)$ of vertices is a dominating set of $G$ if every vertex in $V$ is either in $D$ or adjacent to a vertex of $D$. A dominating set $D$ of $G$ is minimal if $D$ does not properly contain a dominating set of $D$. The vertex set ${b, e, i}$ is a minimal dominating set in the graph in Fig. 5.9. A dominating set $D$ of $G$ is minimum if no other dominating set has fewer vertices than $D$. The cardinality of a minimum dominating set of $G$ is called the domination number of $G$ and denoted by $\gamma(G)$. For the graph in Fig. $5.9 \gamma(G)=2$ and the vertex set ${b, f}$ is a minimum dominating set. We say a vertex in a dominating set dominates itself and all of its neighbors.

The government plans to establish fire stations in a new city in such a way that a locality or one of its neighbor localities will have a fire station. In a graph model of the city, where each vertex represents a locality and each edge represents the neighborhood of two localities, a dominating set gives a feasible solution for the locations of fire stations. If the government wishes to minimize the number of fire stations for budget constraint, a minimum dominating set gives a feasible solution.

Domination of vertices has been studied extensively due its practical applications in scenarios described above [4].

Domination number has a relation with diameter of a graph as we see in the following lemma.

Lemma 5.4.1 Let $G$ be a connected simple graph, $\gamma(G)$ be the domination number of $G$, and $\operatorname{diam}(G)$ be the diameter of $G$. Then
$$\gamma(G) \geq\left\lceil\frac{\operatorname{diam}(G)+1}{3}\right\rceil .$$
Proof Let $x$ and $y$ be two vertices of $G$ such that $d(x, y)=\operatorname{diam}(G)=k$, and let $P=u_0(=x), u_1, \ldots, u_k(=y)$ be a path of length $k$ in $G$ from $x$ to $y$. Let $D$ be a domination set of $G$. We now prove that each vertex in $D$ can dominate at most three vertices on $P$. Let $u$ be a vertex in $D$. If $u$ is on $P, u$ can dominate at most three vertices on $P: v$ itself and its (at most) two neighbors. If $u$ is not on $P, u$ can also dominate at most three vertices on $P$ and those vertices must be consecutive on $P$; otherwise, there would exist a path between $x$ and $y$ shorter than $P$, a contradiction to the definition of $\operatorname{diam}(G)$. Therefore, each vertex in $D$ dominates at most three vertices on $P$.

Since the number of vertices on $P$ is $k+1=\operatorname{diam}(G)+1, \gamma(G) \geq\left\lceil\frac{\operatorname{diam}(G)+1}{3}\right\rceil$.

数学代写|图论作业代写Graph Theory代考|Factor of a Graphraceful Labeling

A factor of a graph $G$ is a spanning subgraph of $G$. A $k$-factor is a spanning $k$-regular subgraph. Clearly, a 1-factor is a perfect matching and exists only for graphs with an even number of vertices. A 2 -factor of $G$ is a disjoint union of cycles of $G$ if the 2 -factor is not connected; a connected 2-factor is a Hamiltonian cycle.

We now know Tutte’s condition [8] for 1-factor. A connected component $H$ of a graph is an odd component if $H$ has odd number of vertices. We denote by $o c(G)$ the number of odd components in a graph $G$. The following theorem is from Tutte [8].
Theorem 5.5.1 A graph $G$ has a l-factor if and only if oc $(G-S) \leq|S|$ for every $S \subseteq V(G)$.

If we delete the vertex $x$ from the graph in Fig.5.11(a), then we get two odd components. Taking $S={x}$, the graph violates the condition in Theorem 5.5.1, and hence it does not have a 1-factor. However, the graph in Fig. 5.11(b) satisfies the condition in Theorem 5.5.1 and it has a 1-factor as shown by thick edges.

We now see an application of Theorem 5.5.1 in the proof of Theorem 5.5.2 (due to Chungphaisan [9]) which gives a necessary and sufficient condition for a tree to have a 1 -factor. The proof presented here is due to Amahashi $[10,11]$.

Theorem 5.5.2 A tree $T$ of even order has a 1 -factor if and only if $o c(T-v)=1$ for every vertex $v$ of $T$.

Proof Assume that $T$ has a 1-factor $F$. Then for every vertex $v$ of $T$, let $w$ be the vertex of $T$ joined to $v$ by an edge of $F$, as illustrated in Fig. 5.12(a). It follows that the component of $T-v$ containing $w$ is odd, and all the other components of $T-v$ are even. Hence $o c(T-v)=1$. Suppose that $o c(T-v)=1$ for every $v \in V(T)$. It is obvious that for each edge $e$ of $T, T-e$ has exactly two components, and both of them are simultaneously odd or even. Define a set $F$ of edges of $T$ as follows: $F={e \in E(T): o c(T-e)=2}$. For every vertex $v$ of $T$, there exists exactly one edge $e$ that is incident with $v$ and satisfies $o c(T-e)=2$ since $T-v$ has exactly one odd component, where $e$ is the edge joining $v$ to this odd component. (See Fig.5.12(b).) Therefore $e$ is an edge of $F$, and thus $F$ is a 1-factor of $G$.

图论代考

数学代写|图论作业代写Graph Theory代考|Dominating Set

$$\gamma(G) \geq\left\lceil\frac{\operatorname{diam}(G)+1}{3}\right\rceil .$$

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|MATH392

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|Distances in Trees and Graphs

In this section, we study some distance parameters of graphs and trees. If $G$ has a $u, v$-path, then the distance from $u$ to $v$ is the length of a shortest $u, v$-path. The distance from $u$ to $v$ in $G$ is denoted by $d_G(u, v)$ or simply by $d(u, v)$. For the graph in Fig. 4.10, $d(a, j)=5$ although there is a path of length 8 between $a$ and $j$. If $G$ has no $u, v$-path then $d(u, v)=\infty$. The diameter of $G$ is the longest distance among the distances of all pair of vertices in $G$. The graph in Fig. 4.10 has diameter 6. The eccentricity of a vertex $u$ in $G$ is $\max {v \in V(G)} d(u, v)$ and denoted by $\epsilon(u)$. Eccentricities of all vertices of the graph in Fig. 4.10 are shown in the figure. The radius of a graph is $\min {u \in V(G)} \epsilon(u)$. The center of a graph $G$ is the subgraph of $G$ induced by vertices of minimum ecentricity. The maximum of the vertex eccentricities is equal to the diameter. The radius and the diameter of the graph in Fig. 4.10 is 3 and 6 respectively. The vertex $h$ is the center of the graph in Fig. 4.10. The diameter and the radius of a disconnected graph are infinite.

Since there is only one path between any two vertices in a tree, computing the distance parameters for trees is not difficult. We have the following lemma on the center of a tree.
Lemma 4.6.1 The center of a tree is a vertex or an edge.
Proof We use induction on the number of vertices in a tree $T$. If $n \leq 2$ then the entire tree is the center of the tree. Assume that $n \geq 3$ and the claim holds for any tree with less than $n$ vertices. Let $T^{\prime}$ be the graph obtained from $T$ by deleting all leaves of $T . T^{\prime}$ has at least one vertex, since $T$ has a non-leaf vertex as $n \geq 3$. Clearly $T^{\prime}$ is connected and has no cycle, and hence $T^{\prime}$ be a tree with less than $n$ vertices. By induction hypothesis the center of $T^{\prime}$ is a vertex or an edge. To complete the proof, we now show that $T$ and $T^{\prime}$ have the same center. Let $v$ be a vertex in $T$. Every vertex $u$ in $T$ which is at maximum distance from $v$ is a leaf. Since all leaves of $T$ have been deleted to obtain $T^{\prime}$ and any path between two non-leaf vertices in $T$ does not contain a leaf, $\epsilon_{T^{\prime}}(u)=\epsilon_T(u)-1$ for every vertex $u$ in $T^{\prime}$. Furthermore, the ecentricity of a leaf is greater than the ecentricity of its neighbor in $T$. Hence the In this section, we study some distance parameters of graphs and trees. If $G$ has a $u, v$-path, then the distance from $u$ to $v$ is the length of a shortest $u, v$-path. The distance from $u$ to $v$ in $G$ is denoted by $d_G(u, v)$ or simply by $d(u, v)$. For the graph in Fig. 4.10, $d(a, j)=5$ although there is a path of length 8 between $a$ and $j$. If $G$ has no $u, v$-path then $d(u, v)=\infty$. The diameter of $G$ is the longest distance among the distances of all pair of vertices in $G$. The graph in Fig. 4.10 has diameter 6. The eccentricity of a vertex $u$ in $G$ is $\max {v \in V(G)} d(u, v)$ and denoted by $\epsilon(u)$. Eccentricities of all vertices of the graph in Fig. 4.10 are shown in the figure. The radius of a graph is $\min {u \in V(G)} \epsilon(u)$. The center of a graph $G$ is the subgraph of $G$ induced by vertices of minimum ecentricity. The maximum of the vertex eccentricities is equal to the diameter. The radius and the diameter of the graph in Fig. 4.10 is 3 and 6 respectively. The vertex $h$ is the center of the graph in Fig. 4.10. The diameter and the radius of a disconnected graph are infinite.

Since there is only one path between any two vertices in a tree, computing the distance parameters for trees is not difficult. We have the following lemma on the center of a tree.

数学代写|图论作业代写Graph Theory代考|Graceful Labeling

A graceful labeling of a simple graph $G$, with $n$ vertices and $m$ edges, is a one-toone mapping $f$ of the vertex set $V$ into the set ${0,1,2, \ldots, m}$, such that distinct vertices receive distinct numbers and $f$ satisfies ${|f(u)-f(v)|: u v \in E(G)}=$ ${1,2,3, \ldots, m}$. The absolute difference $|f(u)-f(v)|$ is regarded as the label of the edge $e=(u, v)$ in the graceful labeling. The number received by a vertex in a graceful labeling is regarded as the label of the vertex. A graph $G$ is called graceful if $G$ admits a graceful labeling.

One can easily compute a graceful labeling of a path as follows. Start labeling of vertices (i.e., assigning labels to vertices) at either end. The first vertex is labeled by 0 and the next vertex on the path is labeled by $n-1$, the next vertex is labeled by 1 , the next vertex is labeled by $n-2$, and so on. Figure 4.12 illustrates a graceful labeling of a path. It is not difficult to observe that edges get labels $n-1, n-2, \ldots, 1$.

An interesting pattern of labeling exits for a graceful labeling of a “caterpillar.” A caterpillar is a tree for which deletion of all leaves produces a path. Observe Fig. 4.13 for an interesting pattern of graceful labeling of a caterpillar.

All trees are graceful — is the famous Ringel-Kotzig conjecture which has been the focus of many papers. Graphs of different classes have been proven mathematically to be graceful or nongraceful. All trees with 27 vertices are graceful was shown by Aldred and McKay using a computer program in 1998. Aryabhatta et. al showed that a fairly large class of trees constructed from caterpillars are graceful [7]. A lobster is a tree for which deletion of all leaves produces a caterpillar. Morgan showed that a subclass of lobsters are graceful [8].

图论代考

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|MA57500

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|Hamiltonian Graphs

An Eulerian circuit visits each edge exactly once, but may visit some vertices more than once. In this section, we consider a round trip through a given graph $G$ such that every vertex is visited exactly once. The original question was posed by a well-known Irish mathematician, Sir William Rowan Hamilton.

Let $G$ be a graph. A path in $G$ that includes every vertex of $G$ is called a Hamiltonian path of $G$. A cycle in $G$ that includes every vertex in $G$ is called a Hamiltonian cycle of $G$. If $G$ contains a Hamiltonian cycle, then $G$ is called a Hamiltonian Graph.
Every Hamiltonian cycle of a Hamiltonian graph of $n$ vertices has exactly $n$ vertices and $n$ edges. If the graph is not a cycle, some edges of $G$ are not included in a Hamiltonian Cycle.

Not all graphs are Hamiltonian. For example, the graph in Fig. 3.6(a) is Hamiltonian, since $a, b, c, d, a$ is a Hamiltonian cycle. On the other hand, the graph in Fig. 3.6(b) is not Hamiltonian, since there is no Hamiltonian cycle in this graph. Note that the path $a, b, c, d, e$ is a Hamiltonian path in the graph in Fig. 3.6(b). Thus a natural question is: What is the necessary and sufficient condition for a graph to be Hamiltonian? Clearly a Hamiltonian graph must be connected and cannot be acyclic, but these are not sufficient. The graph in Fig. 3.6(b) is connected and not acyclic but it is not Hamiltonian. The following lemma gives a necessary condition which is not also sufficient.

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

The connectivity $\kappa(G)$ of a connected graph $G$ is the minimum number of vertices whose removal results in a disconnected graph or a single vertex graph $K_1$. A graph $G$ is $k$-connected if $\kappa(G) \geq k$. A separating set or a vertex cut of a connected graph $G$ is a set $S \subset V(G)$ such that $G-S$ has more than one component. If a vertex cut contains exactly one vertex, then we call the vertex cut a cut vertex. If a vertex cut in a 2-connected graph contains exactly two vertices, then we call the two vertices a separation-pair.

The edge connectivity $\kappa^{\prime}(G)$ of a connected graph $G$ is the minimum number of edges whose removal results in a disconnected graph. A graph is $k$-edge-connected if $\kappa^{\prime}(G) \geq k$. A disconnecting set of edges in a connected graph is a set $F \subseteq E(G)$ such that $G-F$ has more than one component. If a disconnecting set contains exactly one edge, it is called a bridge.

For two disjoint subsets $S$ and $T$ of $V(G)$, we denote [ $S, T]$ the set of edges which have one endpoint in $S$ and the other in $T$. An edge cut is an edge set of the form $[S, \bar{S}]$, where $S$ is a nonempty proper subset of $V(G)$ and $\bar{S}$ denotes $V(G)-S$.
We now explore the relationship among the connectivity $\kappa(G)$, the edge connectivity $\kappa^{\prime}(G)$, and the minimum degree $\delta(G)$ of a connected simple graph $G$. In a cycle of three or more vertices $\kappa(G)=\kappa^{\prime}(G)=\delta(G)=2$. For complete graphs of $n \geq 1$ vertices $\kappa(G)=\kappa^{\prime}(G)=\delta(G)=n-1$. For the graph $G$ in Fig.3.8, $\kappa(G)=1$,$\kappa^{\prime}(G)=2$ and $\delta(G)=3$. Whitney in 1932 showed that the following relationship holds [3].

图论代考

数学代写|图论作业代写Graph Theory代考|CS150

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|Union and Intersection of Graphs

Let $G_1=\left(V_1, E_1\right)$ and $G_2=\left(V_2, E_2\right)$ be two graphs. The union of $G_1$ and $G_2$, denoted by $G_1 \cup G_2$, is another graph $G_3=\left(V_3, E_3\right)$, whose vertex set $V_3=V_1 \cup V_2$ and edge set $E_3=E_1 \cup E_2$.

Similarly, the intersection of $G_1$ and $G_2$, denoted by $G_1 \cap G_2$, is another graph $G_4=\left(V_4, E_4\right)$, whose vertex set $V_4=V_1 \cap V_2$ and edge set $E_4=E_1 \cap E_2$.

Figures 2.10(a) and (b) show two graphs $G_1$ and $G_2$, and Figs. 2.10(c) and (d) illustrate their union and intersection, respectively.

Clearly, we can define the union and intersection of more than two graphs in a similar way. These operations on graphs can be used to solve many problems very easily. We now present such an application of these operations on two graphs [2].
Suppose there are $h+g$ people in a party; $h$ of them are hosts and $g$ of them are guests. Each person shakes hands with each other except that no host shakes hands with any other host. The problem is to find the total number of handshakes. As usual, we transform the scenario into a graph problem as follows. We form a graph with $h+g$ vertices; $h$ of them are black vertices, representing the hosts and the other $g$ vertices are white, representing the guests. The edges of the graph represent the handshakes. Thus, there is an edge between every pair of vertices except for that there is no edge between any pair of black vertices. Thus, the problem now is to count the number of edges in the graph thus formed. The graph is illustrated for $h=3$ and $g=4$ in Fig. 2.11(a).

To solve the problem, we note that the graph can be thought of as a union of two graphs: a complete graph $K_g$ and a complete bipartite graph $K_{h, g}$ as illustrated in Fig. 2.11(b). Since there is no common edge between the two graphs, their intersection contains no edges. Thus, the total number of edges in the graph (i.e., the total number of handshakes in the party) is $n(n-1) / 2+m \times n$.

数学代写|图论作业代写Graph Theory代考|Complement of a Graph

The complement of a graph $G=(V, E)$ is another graph $\bar{G}=(V, \bar{E})$ with the same vertex set such that for any pair of distinct vertices $u, v \in V,(u, v) \in \bar{E}$ if and only if $(u, v) \notin E$. We often denote the complement of a graph $G$ by $\bar{G}$. Figure 2.12(b) illustrates the complement of the graph in Fig. 2.12(a). A null graph is the complement of the complete graph with the same number of vertices and vice versa. The following lemma is an interesting observation in terms of the complement of a graph.
Lemma 2.5.1 For any graph of six vertices, $G$ or $\bar{G}$ contains a triangle.
Proof Let $G$ be a graph of six vertices, and let $v$ be a vertex of $G$. Since the total number of neighbors of $v$ in $G$ and $\bar{G}$ is five, $v$ has at least three neighbors either in $G$ or in $\bar{G}$ by the pigeonhole principle. Without loss of generality we can assume that $v$ has three neighbors $x, y$ and $z$ in $G$. If any two of $x, y$, and $z$ are adjacent to each other, then $G$ contains a triangle. If no two of $x, y$, and $z$ are adjacent, then $x$, $y$, and $z$ will form a triangle in $\bar{G}$.

图论代考

数学代写|图论作业代写Graph Theory代考|MATH3020

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|Graphs and Multigraphs

A graph $G$ is a tuple consisting of a finite set $V$ of vertices and a finite set $E$ of edges where each edge is an unordered pair of vertices. The two vertices associated with an edge $e$ are called the end-vertices of $e$. We often denote by $(u, v)$, an edge between two vertices $u$ and $v$. We also denote the set of vertices of a graph $G$ by $V(G)$ and the set of edges of $G$ by $E(G)$. A vertex of a graph is also called as a node of a graph.
We generally draw a graph $G$ by representing each vertex of $G$ by a point or a small circle and each edge of $G$ by a line segment or a curve between its two endvertices. For example, Fig. 2.1 represents a graph $G$ where $V(G)=\left{v_1, v_2, \ldots, v_{11}\right}$ and $E(G)=\left{e_1, e_2, \ldots, e_{17}\right}$. We often denote the number of vertices of a graph $G$ by $n$ and the number of edges of $G$ by $m$; that is, $n=|V(G)|$ and $m=|E(G)|$. We will use these two notations $n$ and $m$ to denote the number of vertices and the number of edges of a graph unless any confusion arises. Thus $n=11$ and $m=17$ for the graph in Fig. 2.1.

A loop is an edge whose end-vertices are the same. Multiple edges are edges with the same pair of end-vertices. If a graph $G$ does not have any loop or multiple edge, then $G$ is called a simple graph; otherwise, it is called a multigraph. The graph in Fig. 2.1 is a simple graph since it has no loop or multiple edge. On the other hand, the graph in Fig. 2.2 contains a loop $e_5$ and two sets of multiple edges $\left{e_2, e_3, e_4\right}$ and $\left{e_6, e_7\right}$. Hence the graph is a multigraph. In the remainder of the book, when we say a graph, we shall mean a simple graph unless there is any possibility of confusion.
We call a graph a directed graph or digraph if each edge is associated with a direction, as illustrated in Fig. 2.3(a). One can consider a directed edge as a one-way street. We thus can think an undirected graph as a graph where each edge is directed in both directions. We deal with digraphs in Chapter 8. We call a graph a weighted graph if a weight is assigned to each vertex or to each edge. Figure 2.3(b) illustrates an edge-weighted graph where a weight is assigned to each edge.

Let $e=(u, v)$ be an edge of a graph $G$. Then the two vertices $u$ and $v$ are said to be adjacent in $G$, and the edge $e$ is said to be incident to the vertices $u$ and $v$. The vertex $u$ is also called a neighbor of $v$ in $G$ and vice versa. In the graph in Fig.2.1, the vertices $v_1$ and $v_3$ are adjacent; the edge $e_1$ is incident to the vertices $v_1$ and $v_3$. The neighbors of the vertex $v_1$ in $G$ are $v_2, v_3 v_6, v_9$, and $v_{11}$.

The degree of a vertex $v$ in a graph $G$, denoted by $\operatorname{deg}(v)$ or $d(v)$, is the number of edges incident to $v$ in $G$, with each loop at $v$ counted twice. The degree of the vertex $v_1$ in the graph of Fig. 2.1 is 5. Similarly, the degree of the vertex $v_5$ in the graph of Fig. 2.2 is also 5 .

Since the degree of a vertex counts its incident edges, it is obvious that the summation of the degrees of all the vertices in a graph is related to the total number of edges in the graph. In fact the following lemma, popularly known as the “Degree-sum Formula,” indicates that summing up the degrees of each vertex of a graph counts each edge of the graph exactly twice.

Lemma 2.2.1 (Degree-sum Formula) Let $G=(V, E)$ be a graph with $m$ edges. Then $\sum_{v \in V} \operatorname{deg}(v)=2 m$.

Proof Every nonloop edge is incident to exactly two distinct vertices of $G$. On the other hand, every loop edge is counted twice in the degree of its incident vertex in $G$. Thus, every edge, whether it is loop or not, contributes a two to the summation of the degrees of the vertices of $G$.

The above lemma, due to Euler (1736), is an essential tool of graph theory and is sometimes refer to as the “First Theorem of Graph Theory” or the “Handshaking Lemma.” It implies that if some people shake hands, then the total number of hands shaken must be even since each handshake involves exactly two hands. The following corollary is immediate from the degree-sum formula.

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数学代写|图论作业代写Graph Theory代考|Subgraphs

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数学代写|图论作业代写Graph Theory代考|Subgraphs

Let $H$ be a graph and $n \geqslant|H|$. How many edges will suffice to force an $H$ subgraph in any graph on $n$ vertices, no matter how these edges are arranged? Or, to rephrase the problem: which is the greatest possible number of edges that a graph on $n$ vertices can have without containing a copy of $H$ as a subgraph? What will such a graph look like? Will it be unique?

A graph $G \nsupseteq H$ on $n$ vertices with the largest possible number of edges is called extremal for $n$ and $H$; its number of edges is denoted by $\operatorname{ex}(n, H)$. Clearly, any graph $G$ that is extremal for some $n$ and $H$ will also be edge-maximal with $H \nsubseteq G$. Conversely, though, edge-maximality does not imply extremality: $G$ may well be edge-maximal with $H \nsubseteq G$ while having fewer than $\operatorname{ex}(n, H)$ edges (Fig. 7.1.1).

As a case in point, we consider our problem for $H=K^r$ (with $r>1$ ). A moment’s thought suggests some obvious candidates for extremality here: all complete $(r-1)$-partite graphs are edge-maximal without containing $K^r$. But which among these have the greatest number of edges? Clearly those whose partition sets are as equal as possible, i.e. differ in size by at most 1: if $V_1, V_2$ are two partition sets with $\left|V_1\right|-\left|V_2\right| \geqslant 2$, we may increase the number of edges in our complete $(r-1)$-partite graph by moving a vertex from $V_1$ to $V_2$.

The unique complete $(r-1)$-partite graphs on $n \geqslant r-1$ vertices whose partition sets differ in size by at most 1 are called Turán graphs; we denote them by $T^{r-1}(n)$ and their number of edges by $t_{r-1}(n)$ (Fig. 7.1.2). For $n<r-1$ we shall formally continue to use these definitions, with the proviso that – contrary to our usual terminologythe partition sets may now be empty; then, clearly, $T^{r-1}(n)=K^n$ for all $n \leqslant r-1$.

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|Circulations

$$\vec{E}:={(e, x, y) \mid e \in E ; x, y \in V ; e=x y} .$$

$$\bar{F}:={\bar{e} \mid \vec{e} \in \vec{F}}$$

$$\vec{F}(X, Y):={(e, x, y) \in \vec{F} \mid x \in X ; y \in Y ; x \neq y},$$

$$\vec{F}(x):=\vec{F}(x, V)=\vec{F}({x}, \overline{{x}}) .$$

$$f(X, Y):=\sum_{\vec{e} \in \vec{E}(X, Y)} f(\vec{e})$$