## 数学代写|图论作业代写Graph Theory代考|MATH 360

2022年7月11日

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## 数学代写|图论作业代写Graph Theory代考|Long Induced Paths

Our main result is the following, which settles this problem and gives an asymptotically optimal result for the size of a largest induced path in $G(n, p)$.

Theorem 1. For any $\epsilon>0$ there is $d_{0}$ such that whp $G(n, p)$ contains an induced path of length at least $(2-\epsilon) \frac{n}{d} \log d$ whenever $d_{0} \leq d=p n=o(n)$.

For the sake of generality, we state our result for a wide range of functions $d=$ $d(n)$. However, we remark that the most interesting case is when $d$ is a sufficiently large constant. In fact, for dense graphs, when $d \geq n^{1 / 2} \log ^{2} n$, more precise results are already known (cf. $[6,19])$.

Some of the earlier results $[6,11,16]$ are phrased in terms of induced cycles (holes). Using a simple sprinkling argument, one can see that aiming for a cycle instead of a path does not make the problem any harder.

We briefly explain our strategy. Roughly speaking, the idea is to find a long induced path in two steps. First, we find many disjoint paths of some chosen length $L$, such that the subgraph consisting of their union is induced. To achieve this, we generalize a recent result of Cooley, Draganić, Kang and Sudakov.who obtained large induced matchings. We will discuss this further in Sect. 3 . Assuming now we can find such an induced linear forest $F$, the aim is to connect almost all of the small paths into one long induced path, using a few additional vertices. As a “reservoir” for these connecting vertices, we find a large independent set $I$ which is disjoint from $F$. To model the connecting step, we give each path in $F$ a direction, and define an auxiliary digraph whose vertices are the paths, and two paths $\left(P_{1}, P_{2}\right)$ form an edge if there exists a “connecting” vertex $a \in I$ that has some edge to the last $\epsilon L$ vertices of $P_{1}$ and some edge to the first $\epsilon L$ vertices of $P_{2}$, but no edge to the rest of $F$. Our goal is to find an almost spanning path in this auxiliary digraph. Observe that this will provide us with a path in $G(n, p)$ of length roughly $|F|$. The intuition is that the auxiliary digraph behaves quite randomly, which gives us hope that, even though it is very sparse, we can find an almost spanning path.

## 数学代写|图论作业代写Graph Theory代考|Induced Forests with Small Components

As outlined above, in the first step of our argument, we seek an induced linear forest whose components are reasonably long paths. For this, we generalize a recent result of Cooley, Draganić, Kang and Sudakov [3]. They proved that whp $G(n, p)$ contains an induced matching with $\sim 2 \log {q}(n p)$ vertices, which is asymptotically best possible. They also anticipated that using a similar approach one can probably obtain induced forests with larger, but bounded components. As a by-product, we confirm this. To state our result, we need the following definition. For a given graph $T$, a $T$-matching is a graph whose components are all isomorphic to $T$. Hence, a $K{2}$-matching is simply a matching, and the following for $T=K_{2}$ implies the main result of [3].

Theorem 2. For any $\epsilon>0$ and tree $T$, there exists $d_{0}>0$ such that whp the order of the largest induced $T$-matching in $G(n, p)$ is $(2 \pm \epsilon) \log {q}(n p)$, where $q=\frac{1}{1-p}$, whenever $\frac{d{0}}{n} \leq p \leq 0.99$

We use the same approach as in [3], which goes back to the work of Frieze [10] (see also $[1,20]$ ). The basic idea is as follows. Suppose we have a random variable $X$ and want to show that whp, $X \geq b-t$, where $b$ is some “target” value and $t$ a small error. For many natural variables, we know that $X$ is “concentrated”, say $\mathbb{P}[|X-\mathbb{E}[X]| \geq t / 2]<\rho$ for some small $\rho$. This is the case for instance when $X$ is determined by many independent random choices, each of which has a small effect. However, it might be difficult to estimate $\mathbb{E}[X]$ well enough. But if we know in addition that $\mathbb{P}[X \geq b] \geq \rho$, then we can combine both estimates to $\mathbb{P}[X \geq b]>\mathbb{P}[X \geq \mathbb{E}[X]+t / 2]$, which clearly implies that $b \leq$ $\mathbb{E}[X]+t / 2$. Applying now the other side of the concentration inequality, we infer $\mathbb{P}[X \leq b-t] \leq \mathbb{P}[X \leq \mathbb{E}[X]-t / 2]<\rho$, as desired.

## 数学代写|图论作业代写Graph Theory代考|Induced Forests with Small Components

$\mathbb{P}[|X-\mathbb{E}[X]| \geq t / 2]<\rho$ 对于一些小 $\rho$. 例如，当 $X$ 是由许多独立的随机选择决定 的，每个选择的影响很小。不过估计很难 $\mathbb{E}[X]$ 足够好。但是如果我们还知道 $\mathbb{P}[X \geq b] \geq \rho$ ，那么我们可以将两个估计结合起来 $\mathbb{P}[X \geq b]>\mathbb{P}[X \geq \mathbb{E}[X]+t / 2]$ ，这清楚地表明 $b \leq \mathbb{E}[X]+t / 2$. 现在应用集中不 等式的另一面，我们推断 $\mathbb{P}[X \leq b-t] \leq \mathbb{P}[X \leq \mathbb{E}[X]-t / 2]<\rho$ ，如预期的。

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