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# 数学代写|随机图论代考Random Graph Theory代写|MAST30011 Spanning Subgraphs

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## 数学代写|随机图论代考Random Graph Theory代写|Perfect Matchings

Before we move to the problem of the existence of a perfect matching, i.e., a collection of independent edges covering all of the vertices of a graph, in our main object of study, the random graph $\mathbb{G}{n, p}$, we will analyse the same problem in a random bipartite graph. This problem is much simpler than the respective one for $\mathbb{G}{n, p}$, but provides a general approach to finding a perfect matching in a random graph.
Bipartite Graphs
Let $\mathbb{G}{n, n, p}$ be the random bipartite graph with vertex bi-partition $V=(A, B), A=$ $[1, n], B=[n+1,2 n]$ in which each of the $n^2$ possible edges appears independently with probability $p$. The following theorem was first proved by Erdôs and Rényi [290] Theorem 6.1. Let $\omega=\omega(n), c>0$ be a constant, and $p=\frac{\log n+\omega}{n}$. Then $$\lim {n \rightarrow \infty} \mathbb{P}\left(\mathbb{G}{n, n, p} \text { has a perfect matching }\right)= \begin{cases}0 & \text { if } \omega \rightarrow-\infty \ e^{-2 e^{-c}} & \text { if } \omega \rightarrow c \ 1 & \text { if } \omega \rightarrow \infty .\end{cases}$$ Moreover, $$\lim {n \rightarrow \infty} \mathbb{P}\left(\mathbb{G}{n, n, p} \text { has a perfect matching }\right)=\lim {n \rightarrow \infty} \mathbb{P}\left(\delta\left(\mathbb{G}_{n, n, p}\right) \geq 1\right) .$$
Proof. We will use Hall’s condition for the existence of a perfect matching in a bipartite graph. It states that a bipartite graph contains a perfect matching if and only if the following condition is satisfied:
$$\forall S \subseteq A,|N(S)| \geq|S|,$$
where for a set of vertices $S, N(S)$ denotes the set of neighbors of $S$. It is convenient to replace (6.1) by
\begin{aligned} \forall S \subseteq A,|S| \leq \frac{n}{2}, \quad|N(S)| \geq|S| \ \forall T \subseteq B,|T| \leq \frac{n}{2}, \quad|N(T)| \geq|T| \end{aligned}

## 数学代写|随机图论代考Random Graph Theory代写|Non-Bipartite Graphs

We now consider $\mathbb{G}{n, p}$. We could try to replace Hall’s theorem by Tutte’s theorem. A proof along these lines was given by Erdôs and Rényi [291]. We can however get away with a simpler approach based on simple expansion properties of $\mathbb{G}{n, p}$. The proof here can be traced back to Bollobás and Frieze [147].
Theorem 6.2. Let $\omega=\omega(n), c>0$ be a constant, and let $p=\frac{\log n+c_n}{n}$. Then
$$\lim {\substack{n \rightarrow \infty \ n \text { even }}} \mathbb{P}\left(\mathbb{G}{n, p} \text { has a perfect matching }\right)= \begin{cases}0 & \text { if } c_n \rightarrow-\infty \ e^{-e^{-c}} & \text { if } c_n \rightarrow c \ 1 & \text { if } c_n \rightarrow \infty\end{cases}$$
Moreover,
$$\lim {n \rightarrow \infty} \mathbb{P}\left(\mathbb{G}{n, p} \text { has a perfect matching }\right)=\lim {n \rightarrow \infty} \mathbb{P}\left(\delta\left(\mathbb{G}{n, p}\right) \geq 1\right)$$
Proof. We will for convenience only consider the case where $c_n=\omega \rightarrow \infty$ and $\omega=o(\log n)$. If $c_n \rightarrow-\infty$ then there are isolated vertices, w.h.p. and our proof can easily be modified to handle the case $c_n \rightarrow c$.

Our combinatorial tool that replaces Tutte’s theorem is the following: We say that a matching $M$ isolates a vertex $v$ if no edge of $M$ contains $v$.
For a graph $G$ we let
$$\mu(G)=\max {|M|: M \text { is a matching in } G} .$$
Let $G=(V, E)$ be a graph without a perfect matching i.e. $\mu(G)<\lfloor|V| / 2\rfloor$. Fix $v \in V$ and suppose that $M$ is a maximum matching that isolates $v$. Let $S_0(v, M)=$ ${u \neq v: M$ isolates $u}$. If $u \in S_0(v, M)$ and $e={x, y} \in M$ and $f={u, x} \in E$ then flipping $e, f$ replaces $M$ by $M^{\prime}=M+f-e$. Here $e$ is flipped-out. Note that $y \in S_0\left(v, M^{\prime}\right)$.
Now fix a maximum matching $M$ that isolates $v$ and let
$$A(v, M)=\bigcup_{M^{\prime}} S_0\left(v, M^{\prime}\right)$$
where we take the union over $M^{\prime}$ obtained from $M$ by a sequence of flips.

## 数学代写|随机图论代考Random Graph Theory代写|Perfect Matchings

$$\lim n \rightarrow \infty \mathbb{P}(\mathbb{G} n, n, p \text { has a perfect matching })=\lim n \rightarrow \infty \mathbb{P}\left(\delta\left(\mathbb{G}{n, n, p}\right) \geq 1\right)$$ 证明。我们将使用 Hall 的条件来确定二部图中存在完美匹配。它指出当且仅当满足以下条件时，二分图包含完美匹配: $$\forall S \subseteq A,|N(S)| \geq|S|,$$ 哪里有一组顶点 $S, N(S)$ 表示的邻居集 $S$. 将 (6.1) 替换为 $$\forall S \subseteq A,|S| \leq \frac{n}{2}, \quad|N(S)| \geq|S| \forall T \subseteq B,|T| \leq \frac{n}{2}, \quad|N(T)| \geq|T|$$ 我们现在考虑 $\mathbb{G} n, p$. 我们可以尝试用图特定理代替霍尔定理。Erdôs 和 Rényi [291] 给出了沿差这些思路的证明。然而，我们可 以摆脱基于简单扩展属性的更简单方法 $\mathbb{G} n, p$. 这里的证明可以追湖到 Bollobás 和 Frieze [147]。 定理 6.2。让 $\omega=\omega(n), c>0$ 是一个常数，让 $p=\frac{\log n+c_n}{n}$. 然后 $\lim n \rightarrow \infty n$ even $\mathbb{P}(\mathbb{G} n, p$ has a perfect matching $)=\left{\begin{array}{lll}0 & \text { if } c_n \rightarrow-\infty e^{-e^{-c}} & \text { if } c_n \rightarrow c 1\end{array}\right.$ if $c_n \rightarrow \infty$ 而且， $$\lim n \rightarrow \infty \mathbb{P}(\mathbb{G} n, p \text { has a perfect matching })=\lim n \rightarrow \infty \mathbb{P}(\delta(\mathbb{G} n, p) \geq 1)$$ 证明。

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