On Packing Connectors

Given an undirected graphG=(V, E) and a partition {S, T} ofV, anS−Tconnector is a set of edgesF⊆Esuch that every component of the subgraph (V, F) intersects bothSandT. We show thatGhaskedge-disjointS-Tconnectors if and only if |δG(V1)∪…∪δG(Vt)|⩾ktfor every collection {V1, …, Vt} of disjoint nonempty subsets ofSand for every such collection of subsets ofT. This is a common generalization of a theorem of Tutte and Nash-Williams on disjoint spanning trees and a theorem of König on disjoint edge covers in a bipartite graph.