The Minimal Size of a Graph with Given Generalized \(3\)-Edge-Connectivity

Abstract

For \(S \subseteq V(G)\) and \(|S| \geq 2\), let \(\lambda(S)\) denote the maximum number of edge-disjoint trees connecting \(S\) in \(G\). For an integer \(k\) with \(2 \leq k \leq n\), the generalized \(k\)-edge-connectivity \(\lambda_k(G)\) of \(G\) is defined as \(\lambda_k(G) = \min\{\lambda(S) : S \subseteq V(G) \text{ and } |S| = k\}\). Note that when \(|S| = 2\), \(\lambda_2(G)\) coincides with the standard \emph{edge-connectivity} \(\lambda(G)\) of \(G\). In this paper, we characterize graphs of order \(n\) such that \(\lambda_n(G) = n – 3\). Furthermore, we determine the minimal number of edges of a graph \(G\) of order \(n\) with \(\lambda_3(G) = 1, n – 3, n – 2\) and establish a sharp lower bound for \(2 \leq \lambda_3(G) \leq n – 4\).