Bounds on the Maximum Number of Edge-disjoint Steiner Trees of a Graph

Tutte and Nash-Williams, independently, gave necessary and sufficient conditions for a connected graph to have at least t edgedisjoint spanning trees. Gusfield introduced the concept of edgetoughness η(G) of a connected graph G, defined as the minimum |S|/(ω(G − S) − 1) taken over all edge-disconnecting sets S of G, where ω(G−S) is the number of connected components of G−S. If a graph has edge-toughness η(G), Tutte and Nash-Williams’s theorem says that the maximum number of edge-disjoint spanning trees of a graphs is given by bη(G)c. Kundu used this result to show that a graph with edge-connectivity λ(G) has at least bλ(G)/2c edgedisjoint spanning trees. In this paper we investigate to which extent the above results can be generalized to a graph G = (V,E) with a distinguished subset of vertices K. We obtain lower bounds for the maximum number of edge-disjoint Steiner trees of G (minimal trees of G containing K) in terms of λK(G), the K-edge-connectivity of G (defined as the minimum number of edges whose removal disconnects K). In [3] we introduced the K-edge-toughness of a graph, ηK(G) (which coincides with η(G) when K = V ). We extent some of the properties of the edge-toughness of a graph to the K-edge-toughness and we show by mean of a counterexample that the maximum number of disjoint Steiner trees can be less than bηK(G)c when K 6= V . We conclude with some conjectures regarding these bounds.