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 T1 and T2 are said to be internally disjoint if E(T1) ∩ E(T2) = ∅ and V (T1) ∩ V (T2) = S, and edge-disjoint if E(T1) ∩ E(T2) = ∅. The generalized local connectivity κG(S) (generalized local edge-connectivity λG(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 κk(G) = min{κG (S) | S ⊆ V (G), |S| = k} (λk(G) = min{λG(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 κk(G) = t (λk(G) = t, respectively), where 3 ≤ k ≤ n and 1≤t≤n-⌈k2⌉1 \le t \le n - \left\lceil {{k \over 2}} \right\rceil. 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.