Edge-Disjoint Steiner Trees and Connectors in Graphs

Kriesell (J Comb Theory Ser B 88:53-65, 2003) proposed Conjecture 1: If S subset of V (G) is 2k-edge-connected in a graph G, then G contains k edge-disjoint S-Steiner trees. West and Wu (J Comb Theory Ser B 102:186-205, 1961) posed Conjecture 2: If S subset of V (G) is 3k-edge-connected in a graph G, then G contains k edge-disjoint S connectors, which is an analogue for S-connectors of Kriesell's Conjecture. This paper shows If |V(G) - S| <= k, then Conjecture 1 is true and if |V(G) - S| <= 2k, then Conjecture 2 is true. This paper also investigate the validity of two conjectures with certain additional conditions of |V(G) - S| or |S|.