随机和伪随机图的弹性泛圈性
Resilient pancyclicity of random and pseudo-random graphs
摘要：
一个在$n$顶点上的图$G$是\textit{pancyclic}，如果它包含所有$3 \leq t \leq n$的长度为$t$的圈。本文证明了对任意固定的$ε>0$，具有$p(n)\gg n^{-1/2}$的随机图$G(n,p)$几乎必然具有如下弹性性质.若$H$是$G$的一个子图，且其最大度至多为$(1/2 - ε)np$，则$G-H$是泛圈的.事实上，我们证明了一个更一般的结果，即如果$p \gg n^{-1+1/(l-1)}$对于某个整数$l \geq 3$，则对于任何$ε>0$，渐近几乎必然$G(n,p)$的每个最小度大于$(1/2+ε)np$的子图包含长度为$t$的圈，对于所有$l \leq t \leq n$。这些结果在两个方面是紧密的。首先，$p$的条件基本上不能放松。第二，不可能改进最小度假设中的常数1/2。我们也证明了相应的结果，伪随机图。
英文摘要：
A graph $G$ on $n$ vertices is \textit{pancyclic} if it contains cycles of length $t$ for all $3 \leq t \leq n$. In this paper we prove that for any fixed $ε>0$, the random graph $G(n,p)$ with $p(n)\gg n^{-1/2}$ asymptotically almost surely has the following resilience property. If $H$ is a subgraph of $G$ with maximum degree at most $(1/2 - ε)np$ then $G-H$ is pancyclic. In fact, we prove a more general result which says that if $p \gg n^{-1+1/(l-1)}$ for some integer $l \geq 3$ then for any $ε>0$, asymptotically almost surely every subgraph of $G(n,p)$ with minimum degree greater than $(1/2+ε)np$ contains cycles of length $t$ for all $l \leq t \leq n$. These results are tight in two ways. First, the condition on $p$ essentially cannot be relaxed. Second, it is impossible to improve the constant 1/2 in the assumption for the minimum degree. We also prove corresponding results for pseudo-random graphs.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
