数学代写|图论作业代写Graph Theory代考|MATH 3V03

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数学代写|图论作业代写Graph Theory代考|Hamiltonicity of Graphs Perturbed by a Random Geometric Graph

The theory of randomly perturbed graphs serves to bridge between two classical areas of comhinatorics, namely the area of extremal combinatorics and the area of random graphs. In the case of Hamiltonicity, for instance, a classical result of Dirac asserts that every graph $G$ on $n \geq 3$ vertices with minimum degree $\delta(G) \geq$ $n / 2$ contains a Hamilton cycle. As for random graphs, another classical result (originally due to Kors̉unov) states that the random graph $\bar{G}{n, p}$ is Hamiltonian asymptotically almost surely (abbreviated as a.a.s.; we say that a sequence of events $\left{\mathcal{E}{i}\right}_{i \in \mathbb{N}}$ holds a.a.s. if $\mathbb{P}\left[\mathcal{E}{i}\right] \rightarrow 1$ as $\left.i \rightarrow \infty\right)$ whenever $p \geq(1+\varepsilon) \log n / n$, while a.a.s. it is not even connected if $p \leq(1-\varepsilon) \log n / n$ (here, $G{n, p}$ stands for the binomial random graph on $n$ vertices, where each of the possible edges is added to the graph with probability $p$, independently of all other edges). Bohman, Frieze and Martin [2] were the first to consider the union of a deterministic graph and a random graph, and they proved the following result.

Theorem $1([2])$. For every $\alpha \in(0,1 / 2)$, there exists a constant $C$ such that, if $H$ is an $n$-vertex graph with $\delta(H) \geq \alpha n$ and $p \geq C / n$, then a.a.s. $H \cup G_{n, p}$ is Hamiltonian.

数学代写|图论作业代写Graph Theory代考|Hamiltonicity of Graphs Perturbed

A $d$-dimensional random geometric graph $G^{d}(n, r)$, where $r$ is a positive real number, is a graph with vertex set $V:=[n]$ and edge set $E$ defined as follows. Let $X_{1}, \ldots, X_{n}$ be $n$ independent uniform random variables on $[0,1]^{d}$. Then, let $E:=\left{{i, j}:\left|X_{i}-X_{j}\right| \leq r\right}$, where $|\cdot|$ denotes the Euclidean norm. (The results in this section extend to any $\ell_{p}$ norm, but we consider $p=2$ here for simplicity.)

Hamiltonicity is fairly well-understood in random geometric graphs. In particular, for all $d \geq 1$ there exists a constant $c_{d}$ such that, for all $\varepsilon>0$, if $r \geq(1+\varepsilon)\left(c_{d} \log n / n\right)^{1 / d}$, then a.a.s. $G^{d}(n, r)$ is Hamiltonian, and if $r \leq$ $(1-\varepsilon)\left(c_{d} \log n / n\right)^{1 / d}$, then a.a.s. $G^{d}(n, r)$ is not Hamiltonian $[1,5,12]$. The main result of the first author in this context is the following, which can be seen as an analogue of Theorem 1 for random geometric graphs.

Theorem $2([6])$. For every integer $d \geq 1$ and $\alpha \in(0,1 / 2)$, there exists a constant $C^{\prime}$ such that the following holds. Let $H$ be an $n$-vertex graph with $\delta(H) \geq$ $\alpha n$, and let $r \geq(C / n)^{1 / d}$. Then, a.a.s. $H \cup G^{d}(n, r)$ is Hamiltonian.

Proof (sketch). We choose some constant $C$, which depends on $\alpha$ and $d$ and is sufficiently large so that all subsequent claims hold. We assume that $r=$ $(C / n)^{1 / d}$ (which suffices since, by increasing $r$, we create a sequence of nested graphs).

Partition the hypercube $[0,1]^{d}$ into smaller cubes of side $y$, which we call cells. Two cells $c_{1}$ and $c_{2}$ are friends if there is a cell $c_{3}$ whose boundary intersects those of both $c_{1}$ and $c_{2}$. The value of $y$ is chosen so that, if $c_{1}$ and $c_{2}$ are friends, then all $x_{1} \in c_{1}, x_{2} \in c_{2}$ satisfy $\left|x_{1}-x_{2}\right| \leq r$ (i.e., all vertices in $c_{1}$ will be adjacent to all vertices in $\left.c_{2}\right)$. In particular, $y=\Theta(r)$ and there are $\Theta(n)$ cells. Consider the variables $X_{1}, \ldots, X_{n}$ that assign positions in $[0,1]^{d}$ to the vertices of $H$. A cell is called dense if it contains at least $2 \cdot 5^{d}$ vertices of $H$, and sparse otherwise. Furthermore, for each pair of (not necessarily distinct) vertices $u, v \in V(H)$, we call a cell ${u, v}$-dense if it contains two distinct vertices $w$ and $x$ such that $u w, v x \in E(H)$, and ${u, v}$-sparse otherwise. By Azuma’s inequality (and adjusting the value of $C$ ), we can show that a.a.s. the proportion of sparse cells is an arbitrarily small constant, and similarly the proportion of ${u, v}$-sparse cells is an arbitrarily small constant, for all pairs of vertices $u, v \in V(H)$.

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