Edge-Disjoint Steiner Trees and Connectors in Graphs

Kriesell (J Comb Theory Ser B 88:53–65, 2003) proposed Conjecture 1: If S ⊆ V ( G ) is 2 k -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 ⊆ V ( G ) is 3 k -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 | ≤ 2 k , then Conjecture 2 is true. This paper also investigate the validity of two conjectures with certain additional conditions of | V ( G ) - S | or | S |.