The generalized 3-connectivity of random graphs

The generalized connectivity of a graph $G$ was introduced by Chartrand et al. Let $S$ be a nonempty set of vertices of $G$, and $\kappa(S)$ be defined as the largest number of internally disjoint trees $T_1, T_2, \cdots, T_k$ connecting $S$ in $G$. Then for an integer $r$ with $2 \leq r \leq n$, the {\it generalized $r$-connectivity} $\kappa_r(G)$ of $G$ is the minimum $\kappa(S)$ where $S$ runs over all the $r$-subsets of the vertex set of $G$. Obviously, $\kappa_2(G)=\kappa(G)$, is the vertex connectivity of $G$, and hence the generalized connectivity is a natural generalization of the vertex connectivity. Similarly, let $\lambda(S)$ denote the largest number $k$ of pairwise edge-disjoint trees $T_1, T_2, \ldots, T_k$ connecting $S$ in $G$. Then the {\it generalized $r$-edge-connectivity} $\lambda_r(G)$ of $G$ is defined as the minimum $\lambda(S)$ where $S$ runs over all the $r$-subsets of the vertex set of $G$. Obviously, $\lambda_2(G) = \lambda(G)$. In this paper, we study the generalized 3-connectivity of random graphs and prove that for every fixed integer $k\geq 1$, $$p=\frac{{\log n+(k+1)\log \log n -\log \log \log n}}{n}$$ is a sharp threshold function for the property $\kappa_3(G(n, p)) \geq k$, which could be seen as a counterpart of Bollob\'{a}s and Thomason's result for vertex connectivity. Moreover, we obtain that $\delta (G(n,p)) - 1 = \lambda (G(n,p)) - 1 = \kappa (G(n,p)) - 1 \le {\kappa_3}(G(n,p)) \le {\lambda_3}(G(n,p)) \le \kappa (G(n,p)) = \lambda (G(n,p)) = \delta (G(n,p))$ almost surely holds, which could be seen as a counterpart of Ivchenko's result.