Edge-connectivity of undirected and directed hypergraphs

130 Chapter 8. Concluding remarks
where µ = min{γ, |{sv ∈ A : v /∈ ∩ t i=1Xi}|}.
Let us return to undirected hypergraphs. We have shown in Chapter 5 how matroid-
theoretic tools applied on hypergraphic matroids can be used to prove results like Theorem
5.7 on the decomposition of a hypergraph into k partition-connected sub-hypergraphs. This
result in turn easily implies Theorem 5.12 which gives a sufficient condition for the existence
of k edge-disjoint Steiner trees for W when V − W is stable. Kriesell [53] conjectures that
2k-edge-connectivity in W is sufficient, even in the case when V − W is not necessarily
stable. To date, no constant c is known such that ck-edge-connectivity in W guarantees
the existence of k edge-disjoint Steiner trees.
Another open problem for graphs that is related to partition-connectivity is the con-
structive characterization of (k, l)-partition-connected graphs, i.e. Conjecture 6.7. This
conjecture would follow from Conjecture 6.6 on the constructive characterization of (k, l)-
edge-connected digraphs. Since by Theorem 6.11 there is a similar relation between (k, l)-
partition-connectivity and (k, l)-edge-connectivity for hypergraphs, we might obtain similar
constructive characterizations for hypergraphs as well.
One of the exciting questions in graph connectivity orientation is the solvability of ori-
entation problems with parity constraints. Known results were cited in Chapter 6; a basic
open problem is the characterization of graphs which have a strongly connected orientation
where the in-degree of every node is odd. Of course, similar questions can be asked for
hypergraphs.
A problem where the answer is known for graphs but not for hypergraphs is local edge-
connectivity orientation with symmetric demands. For example, no characterization is
known for hypergraphs which have a k-edge-connected orientation that is at the same time
l-edge-connected in a given subset W (l > k). For graphs, this problem is solved by Theo-
rem 6.4 of Nash-Williams, whose proof makes use of the basic properties of Eulerian graphs;
so it might be interesting to find an appropriate definition of “Eulerian hypergraphs” that
is useful in this context.
One can also extend orientation problems to mixed graphs and hypergraphs. Instances
where we obtain tractable problems are the framework of directed network design with ori-
entation constraints, introduced by Khanna, Naor, and Shepherd (an extended hypergraph
version is described in Theorem 6.14), and Theorem 6.5 of Frank. The latter can be seen
as a special case of the combined augmentation and orientation problem solved in Theorem
7.13. This theorem gives a characterization for a degree-specified augmentation problem;
we do not yet know how to solve the corresponding minimum cardinality problem.