The Minimum Size Of A Graph With Given Tree Connectivity

For a graph G = (V, E) and a set S subset of V of at least two vertices, an S-tree is a such subgraph T of G that is a tree with S subset of V(T). Two S-trees T-1 and T-2 are said to be internally disjoint if E(T-1) boolean AND E(T-2) = empty set and V(T-1) boolean AND V(T-2) = S, and edge-disjoint if E(T-1) boolean AND E(T-2) = empty set. The generalized local connectivity kappa(G) (S) (generalized local edge-connectivity lambda(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 kappa(k)(G) = min{kappa(G)(S) vertical bar S subset of V (G),vertical bar S vertical bar = k} (lambda(k) (G) = min{lambda(G) (S) vertical bar S subset of V (G),vertical bar S vertical bar = 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 kappa(k) (G) = t (lambda((k)G)= t, respectively), where 3 <= k <= n and 1 <= t <= n- inverted right perpendicular k/2 inverted left perpendicular. 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 is an element of {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 is an element of {1, 2, 3, 4} and an upper bound of f(n,k,t) for t is an element of {1, 2, 3}. Moreover, both bounds can be attained.