当前位置: X-MOL 学术Mathematika › 论文详情
Our official English website, www.x-mol.net, welcomes your feedback! (Note: you will need to create a separate account there.)
EMBEDDING SPANNING BOUNDED DEGREE GRAPHS IN RANDOMLY PERTURBED GRAPHS
Mathematika ( IF 0.8 ) Pub Date : 2020-04-01 , DOI: 10.1112/mtk.12005
Julia Böttcher 1 , Richard Montgomery 2 , Olaf Parczyk 3 , Yury Person 3
Affiliation  

We study the model $G_\alpha\cup G(n,p)$ of randomly perturbed dense graphs, where $G_\alpha$ is any $n$-vertex graph with minimum degree at least $\alpha n$ and $G(n,p)$ is the binomial random graph. We introduce a general approach for studying the appearance of spanning subgraphs in this model, which we call assisted absorption. This approach yields simpler proofs of several known results. We also use it to derive the following two new results. For every $\alpha>0$ and $\Delta\ge 5$, and every $n$-vertex graph $F$ with maximum degree at most $\Delta$, we show that if $p=\omega(n^{-2/(\Delta+1)})$ then $G_\alpha \cup G(n,p)$ with high probability contains a copy of $F$. The bound used for $p$ here is lower by a $\log$-factor in comparison to the conjectured threshold for the general appearance of such subgraphs in $G(n,p)$ alone, a typical feature of previous results concerning randomly perturbed dense graphs. We also give the first example of graphs where the appearance threshold in $G_\alpha \cup G(n,p)$ is lower than the appearance threshold in $G(n,p)$ by substantially more than a $\log$-factor. We prove that, for every $k\geq 2$ and $\alpha >0$, there is some $\eta>0$ for which the $k$th power of a Hamilton cycle with high probability appears in $G_\alpha \cup G(n,p)$ when $p=\omega(n^{-1/k-\eta})$. The appearance threshold of the $k$th power of a Hamilton cycle in $G(n,p)$ alone is known to be $n^{-1/k}$, up to a $\log$-term when $k=2$, and exactly for $k>2$.

中文翻译:

在随机扰动图中嵌入跨越有界度图

我们研究随机扰动密集图的模型 $G_\alpha\cup G(n,p)$,其中 $G_\alpha$ 是任何 $n$-顶点图,其最小度至少为 $\alpha n$ 和 $G (n,p)$ 是二项式随机图。我们介绍了一种研究该模型中生成子图外观的通用方法,我们称之为辅助吸收。这种方法产生了几个已知结果的更简单的证明。我们还使用它来推导出以下两个新结果。对于每个 $\alpha>0$ 和 $\Delta\ge 5$,以及每个 $n$-顶点图 $F$ 的最大度数最多为 $\Delta$,我们证明如果 $p=\omega(n^ {-2/(\Delta+1)})$ 然后 $G_\alpha \cup G(n,p)$ 以高概率包含 $F$ 的副本。与仅在 $G(n,p)$ 中此类子图的一般出现的推测阈值相比,此处用于 $p$ 的界限要低一个 $\log$-factor,先前关于随机扰动密集图的结果的典型特征。我们还给出了第一个图的例子,其中 $G_\alpha\cup G(n,p)$ 中的出现阈值比 $G(n,p)$ 中的出现阈值低得多 $\log$ -因素。我们证明,对于每一个 $k\geq 2$ 和 $\alpha >0$,都有一些 $\eta>0$ 使得一个高概率的哈密顿循环的 $k$th 次幂出现在 $G_\alpha 中\cup G(n,p)$ 当 $p=\omega(n^{-1/k-\eta})$ 时。仅在 $G(n,p)$ 中哈密顿循环的 $k$th 次幂的出现阈值已知为 $n^{-1/k}$,当 $ k=2$,正好是 $k>2$。p)$ 比 $G(n,p)$ 中的出现阈值低得多一个 $\log$-factor。我们证明,对于每一个 $k\geq 2$ 和 $\alpha >0$,都有一些 $\eta>0$ 使得一个高概率的哈密顿循环的 $k$th 次幂出现在 $G_\alpha 中\cup G(n,p)$ 当 $p=\omega(n^{-1/k-\eta})$ 时。仅在 $G(n,p)$ 中哈密顿循环的 $k$th 次幂的出现阈值已知为 $n^{-1/k}$,当 $ k=2$,正好是 $k>2$。p)$ 比 $G(n,p)$ 中的出现阈值低得多一个 $\log$-factor。我们证明,对于每一个 $k\geq 2$ 和 $\alpha >0$,都有一些 $\eta>0$ 使得一个高概率的哈密顿循环的 $k$th 次幂出现在 $G_\alpha 中\cup G(n,p)$ 当 $p=\omega(n^{-1/k-\eta})$ 时。仅在 $G(n,p)$ 中哈密顿循环的 $k$th 次幂的出现阈值已知为 $n^{-1/k}$,当 $ k=2$,正好是 $k>2$。
更新日期:2020-04-01
down
wechat
bug