Highly connected hypergraphs containing no two edge-disjoint spanning connected subhypergraphs

We prove that there is no degree of connectivity which will guarantee that a hypergraph contains two edge-disjoint spanning connected subhypergraphs. We also show that Edmonds’ theorem on arc-disjoint branchings cannot be extended to directed hypergraphs. Here we use a definition of a directed hypergraph that naturally generalizes the notion of a directed graph. For standard notation and results on digraphs and hypergraphs we refer to [1] and [2]. A spanning tree of a graph G = (V,E) is a subtree which contains all vertices of G. A graph G = (V,E) is k-edge-connected if and only if there are at least k edges connecting X to V −X for every non empty proper subset X of V . Clearly G is 1-edge-connected if and only if G contains a spanning tree. However, it is not true that every k-edge-connected graph contains k-edge-disjoint spanning trees and hence k-edge-connectivity is not sufficient to ensure that a graph can be decomposed into k edge-disjoint spanning subgraphs. Tutte characterized those graphs which have k edge-disjoint spanning trees. A partition of a set S is a collection of disjoint non empty subsets S1, S2, . . . , St ⊆ S such that S = ⋃t i=1 Si. Theorem 1 (Tutte) [10] A graph G = (V,E) has k edge-disjoint trees if and only if for every partition P = {V1, V2, . . . , Vt} of V , the number of edges in G which connect different sets in P is at least k(t− 1). It is easy to check that Tutte’s theorem implies that every 2k-edge-connected graph can be decomposed into k edge-disjoint spanning subgraphs and we can also use the condition in Theorem 1 to show that 2k is best possible. Tutte’s theorem can be proved in at least two different ways: An out-branching from s in a digraph is a tree which is oriented in such a way that every vertex other than s has precisely one arc coming in. It is easy to see that a graph G has k edgedisjoint spanning trees if and only if it can be oriented as a digraph D so that D ∗Department of Mathematics and Computer Science, University of Southern Denmark, Odense DK-5230, Denmark (email [email protected]). †LaPCS, Université Claude Bernard, Lyon 1, France (email [email protected])