AI 中文总结
本文针对二项随机图G(n,p)，证明其韧性τ(G(n,p))以高概率满足τ(G(n,p))=(n−a)/a+o(1)，并存在序列n_m→∞使得其韧性以高概率小于(n_m−a)/a−1/a。
AI 中文摘要
对于阶数为n的连通非完全图G，其韧性定义为τ(G)=min{|S|/c(G−S):S⊆V(G),c(G−S)>1}，其中c(G−S)表示G−S的连通分支数，α(G)表示G的独立数，韧性的一个基本界为τ(G)≤(n−α(G))/α(G)。令G(n,p)为顶点集为[n]的二项随机图，设a=α(G(n,p))，本文证明，当概率趋近于1时，τ(G(n,p))=(n−a)/a+o(1)；此外，存在序列n_m→∞，使得当概率趋近于1时，τ(G(n_m,p))≤(n_m−a)/a−1/a。
英文摘要
For a connected and non-complete graph $G$ of order $n$, its toughness is defined as \[ τ(G)=\min\bigl\{|S|/c(G-S):S\subseteq V(G),\ c(G-S)>1\bigr\}, \] where $c(G-S)$ denotes the number of components of $G-S$. Let $α(G)$ denote the independence number of $G$. An elementary bound on toughness is $$τ(G)\leq\frac{n-α(G)}{α(G)}.$$ Fix $p\in(0,1)$, and let $G(n,p)$ be the binomial random graph on vertex set $[n]$. Set $a=α(G(n,p))$. In this paper, we mainly prove that \[ τ(G(n,p))\in\left\{\frac{n-a}{a},\frac{n-a-1}{a}\right\} \] with high probability.
