The minimal size of a graph with given generalized 3-edge-connectivity

For $S\subseteq V(G)$ and $|S|\geq 2$, $λ(S)$ is the maximum number of edge-disjoint trees connecting $S$ in $G$. For an integer $k$ with $2\leq k\leq n$, the \emph{generalized $k$-edge-connectivity} $λ_k(G)$ of $G$ is then defined as $λ_k(G)= min\{λ(S) : S\subseteq V(G) \ and \ |S|=k\}$. It is also clear that when $|S|=2$, $λ_2(G)$ is nothing new but the standard edge-connectivity $λ(G)$ of $G$. In this paper, graphs of order $n$ such that $λ_3(G)=n-3$ is characterized. Furthermore, we determine the minimal number of edges of a graph of order $n$ with $λ_3=1,n-3,n-2$ and give a sharp lower bound for $2\leq λ_3\leq n-4$.

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