Edge-connectivity of undirected and directed hypergraphs

Section 7.4. More general requirement functions 127
There exists a hypergraph H = (V, E) with γ hyperedges such that H0+H has an orientation
covering p and dH(v) = m(v) for every v ∈ V if and only if
p m (Z) + (γ − m(X)) + ≤ eH0(F) + (hX(F) + 1)γ (7.24)
Z∈F
for every X ⊆ V and every tree-composition F of X. In addition, H can be chosen so that
m(V )
m(V )
≤ |e| ≤
γ
γ
for every e ∈ E. (7.25)
Proof. The necessity follows from the fact that if F is a tree-composition of X, then
F ′ := F + {V − X} is a regular family,
Z∈F ′ ϱ H0 (Z) ≤ eH0(F ′ ) for any orientation H0
of H0, and
Z∈F ′ ϱ H (Z) ≤ h∅(F ′ )γ −
Z∈F ′ (γ − m(V − Z)) + for any orientation H of a
hypergraph H satisfying the degree specification.
The sufficiency can be proved in essentially the same way as in the proof of Theorem
7.1. We define H ′ 0 similarly, and for X ⊆ V , let
p1(X) := iH0(X) + (γ − m(V − X)) + ,
p2(X) := p(X) + iH0(X) + (γ − m(V − X)) + .
In this case Lemma 6.10 implies that an orientation of H ′ 0 satisfying (7.5)–(7.7) exists if
and only if the polyhedron
{x : V → Q : x(V ) = p1(V ), x(Z) ≥ p1(Z) ∀Z ⊆ V, x(Z) ≥ p2(Z) ∀Z ⊆ V }
has an integral point.
Claim 7.14. The set function p1 is fully supermodular, and the set function p2 is super-
modular on the crossing pairs (X, Y ) for which p1(X) < p2(X) and p1(Y ) < p2(Y ).
Proof. The set function p1 is the sum of two fully supermodular functions (see the proof
of Claim 7.2), so it is fully supermodular. Since p is positively crossing supermodular, p2
is supermodular on the crossing pairs (X, Y ) for which p(X) > 0 and p(Y ) > 0, and these
are exactly the crossing pairs for which p1(X) < p2(X) and p1(Y ) < p2(Y ).
if
Theorem 2.24 implies that an orientation of H ′ 0 satisfying (7.5)–(7.7) exists if and only
p1(V − X) +
p2(Z) ≤ (hX(F) + 1)p1(V )
Z∈F