Packing of Steiner trees and S-connectors in graphs

Nash-Williams and Tutte independently characterized when a graph has k edge-disjoint spanning trees; a consequence is that 2k-edge-connected graphs have k edge-disjoint spanning trees. Kriesell conjectured a more general statement: defining a set S⊆V(G) to be j-edge-connected in G if S lies in a single component of any graph obtained by deleting fewer than j edges from G, he conjectured that if S is 2k-edge-connected in G, then G has k edge-disjoint trees containing S. Lap Chi Lau proved that the conclusion holds whenever S is 24k-edge-connected in G.We improve Lauʼs result by showing that it suffices for S to be 6.5k-edge-connected in G. This and an analogous result for packing stronger objects called “S-connectors” follow from a common generalization of the Tree Packing Theorem and Hakimiʼs criterion for orientations with specified outdegrees. We prove the general theorem using submodular functions and the Matroid Union Theorem.