The Minimum Size of a Graph with Given Tree Connectivity

For a graph $G = (V, E)$ and a set $S ⊆ V$ of at least two vertices, an $S$-tree is a such subgraph $T$ of $G$ that is a tree with $S ⊆ V(T)$. Two $S$-trees $T_1$ and $T_2$ are said to be internally disjoint if $E(T_1) ∩ E(T_2) = ∅$ and $V(T_1) ∩ V(T_2) = S$, and edge-disjoint if $E(T_1) ∩ E(T_2) = ∅$. The generalized local connectivity $κ_G(S)$ (generalized local edge-connectivity $λ_G(S)$, respectively) is the maximum number of internally disjoint (edge-disjoint, respectively) $S$-trees in $G$. For an integer $k$ with $2 ≤ k ≤ n$, the generalized $k$-connectivity (generalized $k$-edge-connectivity, respectively) is defined as $κ_k(G) = min{κ_G(S) | S ⊆ V(G), |S| = k} (λ_k(G) = min{λ_G(S) | S ⊆ V(G), |S| = k}$, respectively). Let $f(n, k, t)$ ($g(n, k, t)$, respectively) be the minimum size of a connected graph $G$ with order $n$ and $κ_k(G) = t$ ($λ_k(G) = t$, respectively), where $3 ≤ k ≤ n$ and \(1≤t≤n-⌈\frac{k}{2}⌉\). For general $k$ and $t$, Li and Mao obtained a lower bound for $g(n, k, t)$ which is tight for the case $k = 3$. We show that the bound also holds for $f(n, k, t)$ and is tight for the case $k = 3$. When t is general, we obtain upper bounds of both $f(n, k, t)$ and $g(n, k, t)$ for $k ∈ {3, 4, 5}$, and all of these bounds can be attained. When $k$ is general, we get an upper bound of $g(n, k, t)$ for $t ∈ {1, 2, 3, 4}$ and an upper bound of $f(n, k, t)$ for $t ∈ {1, 2, 3}$. Moreover, both bounds can be attained.