Steiner tree packing number and tree connectivity

Let S be a set of at least two vertices in a graph G. A subtree T of G is a S-Steiner tree if S subset of V(T). Two S-Steiner trees T-1 and T-2 are edge-disjoint (resp. internally vertex-disjoint) if E(T-1) boolean AND E(T-2) = (sic) (resp. E(T-1) boolean AND E(T-2) = (sic) and V(T-1) boolean AND V(T-2) = S). Let lambda(G)(S) (resp. KG(S)) be the maximum number of edge-disjoint (resp. internally vertex-disjoint) S-Steiner trees in G, and let lambda(k)(G) (resp. kappa(k)(G)) be the minimum lambda(G)(S) (resp. kappa(G)(S)) for S ranges over all k-subset of V(G). Kriesell conjectured that if lambda(G)({x, y}) >= 2k for any x, y is an element of S, then lambda(G)(S) >= k. He proved that the conjecture holds for vertical bar S vertical bar = 3, 4. In this paper, we give a short proof of Kriesell's Conjecture for vertical bar S vertical bar = 3, 4, and also show that lambda(k)(G) >= (left perpendicular)1/k-1 (inverted right perpendicular)kl/2(inverted right perpendicular))(left perpendicular) (rep. kappa(k)(G) >= (left perpendicular)1/k-1 (inverted right perpendicular)kl/2(inverted right perpendicular))(left perpendicular) if lambda(G) >= l in G, where k = 3, 4. Moreover, we also study the relation between kappa(k)(L(G)) and lambda(k)(G), where L(G) is the line graph of G. (C) 2018 Elsevier B.V. All rights reserved.