The minimal size of a graph with generalized connectivity $\kappa_3 = 2$

Abstract

Let $G$ be a nontrivial connected graph of order $n$ and $k$ an integer with $2\leq k\leq n$. For a set $S$ of $k$ vertices of $G$, let $\kappa (S)$ denote the maximum number $\ell$ of edge-disjoint trees $T_1,T_2,...,T_\ell$ in $G$ such that $V(T_i)\cap V(T_j)=S$ for every pair $i,j$ of distinct integers with $1\leq i,j\leq \ell$. Chartrand et al. generalized the concept of connectivity as follows: The $k$-$connectivity$, denoted by $\kappa_k(G)$, of $G$ is defined by $\kappa_k(G)=$min$\{\kappa(S)\}$, where the minimum is taken over all $k$-subsets $S$ of $V(G)$. Thus $\kappa_2(G)=\kappa(G)$, where $\kappa(G)$ is the connectivity of $G$. This paper mainly focuses on the minimal number of edges of a graph $G$ with $\kappa_{3}(G)= 2$. For a graph $G$ of order $v(G)$ and size $e(G)$ with $\kappa_{3}(G)= 2$, we obtain that $e(G)\geq 6/5v(G)$, and the lower bound is sharp by showing a class of examples attaining the lower bound.