The toughness of random graphs

Abstract

For a connected and non-complete graph $G$ of order $n$, its toughness is defined as \[ \tau(G)=\min\bigl\{|S|/c(G-S):S\subseteq V(G),\ c(G-S)>1\bigr\}, \] where $c(G-S)$ denotes the number of components of $G-S$. Let $\alpha(G)$ denote the independence number of $G$. An elementary bound on toughness is $$\tau(G)\leq\frac{n-\alpha(G)}{\alpha(G)}.$$ Fix $p\in(0,1)$, and let $G(n,p)$ be the binomial random graph on vertex set $[n]$. Set $a=\alpha(G(n,p))$. In this paper, we mainly prove that \[ \tau(G(n,p))\in\left\{\frac{n-a}{a},\frac{n-a-1}{a}\right\} \] with high probability.