Edge-connectivity of undirected and directed hypergraphs

90 Chapter 5. Partition-connectivity of hypergraphs
A well-known corollary of Theorem 1.5 of Tutte is that a (2k)-edge-connected graph
always contains k disjoint spanning trees. As a direct extension of this result we can derive
the following:
Corollary 5.8. A (νk)-edge-connected hypergraph H of rank at most ν can be decom-
posed into k partition-connected sub-hypergraphs and hence into k connected spanning sub-
hypergraphs.
Proof. By Theorem 5.7 it suffices to show that H is k-partition-connected. Let F be
a partition of V . By the (νk)-edge-connectivity of H, there are at least νk hyperedges
intersecting both X and V −X for each member X of F. Since one hyperedge may intersect
at most ν members, we obtain that the total number of hyperedges intersecting more than
one member of F is at least (νk)|F|/ν ≥ k(|F| − 1), so H is k-partition-connected.
Note that the proof shows that a bit stronger result is also true: if H is a (νk)-edge-
connected hypergraph of rank at most ν, then after the deletion of any k hyperedges, H
still decomposes into k partition-connected sub-hypergraphs. This kind of connectivity
property is similar to the (k, l)-partition-connectivity of graphs introduced in Chapter 1,
and indeed it is useful to extend this notion to hypergraphs.
5.2.4 (k, l)-partition-connectivity
For non-negative integers k and l, a hypergraph H = (V, E) is called (k, l)-partition-
connected if eH(F) ≥ k(|F|−1)+l for every nontrivial partition F. An equivalent definition
is that no matter how we delete l hyperedges from H, the resulting hypergraph decomposes
into k partition-connected sub-hypergraphs. This may be regarded as a combination of
two different approaches to connectivity: that the hypergraph should be decomposable into
many well-connected components, and that the hypergraph should remain well-connected
even after the deletion of some hyperedges.
A notable difference from the graph case is that while (k, l)-partition-connectivity for k ≤
l is equivalent to (k +l)-edge-connectivity for graphs, the same is not true for hypergraphs.
As a consequence, the case k < l also has some interest.
Note that if |E| = |V | for a hypergraph H = (V, E), then H is (1, 1)-partition-connected
if and only if it is a hypercircuit. Indeed, suppose that iH(X) ≥ |X| for some X ⊂ V . Then
for the partition F := {X}∪{{v} : v ∈ V −X} we have eH(F) ≤ |E|−iH(X) ≤ |V |−|X| ≤
|F| − 1. Conversely, if there is a partition for which eH(F) ≤ |F| − 1, then by assuming
that the strong Hall condition holds we get that |V | = |E| = eH(F) +
Z∈F iH(Z) ≤
|F| − 1 +
Z∈F (|Z| − 1) ≤ |V | − 1, a contradiction.