Modelos clásicos de la localización industrial

Proof. Let γ, k, and r be given. We applyLemma 44 to obtain n₀. Let H be a k-uniform hypergraph on n ≥ n0 vertices with minimum r-degree δr(H) ≥ (1/2 + 2γ) _k−rn
. Then using Lemma 44 we can remove a matching M of size γkn/k from H. According to

the assumption, the remaining hypergraph H[V \ V (M )] contains a matching M0 such that, W , the set of the uncovered vertices has size at most γ2k_{n ≥ |W | ∈ kN. But due} to Lemma 44 there is a matching covering exactly those vertices in V (M ) ∪ W , which together with M0 forms a perfect matching of H.

4.2. General upper bounds for k-uniform hypergraphs

In this section we prove Theorem 6 and Theorem 7. For this we verify general upper bounds on the minimum r-degree, which guarantee the existence of a perfect matching and nearly perfect matching in a k-uniform hypergraphs H. This will be derived from a corresponding result on nearly perfect matchings for k-uniform, k-partite hypergraphs. Here the minimum r-degree δ_r(H) of a k-uniform, k-partite hypergraph with vertex partition V₀∪ . . . ˙˙ ∪V_k−1 is min deg(v_i1, . . . , vir), where the minimum runs over all index

sets {i₁, . . . , ir} ∈ {0,...,k−1}_r
and all r-sets of vertices vij ∈ Vij for j = 1, . . . , r.

Theorem 47. Let H be a k-uniform, k-partite hypergraph on the partition classes

V0, . . . Vk−1 each of size |Vi| = n and suppose the minimum r-degree o f H is

δr(H) >

k − r k n

k−r_{+ kn}k−r−1_.

Then H contains a matching covering all but (r − 1)k vertices. In particular, for r = 1 the matching is perfect.

Let H be a k-uniform, k-partite hypergraph on the partition classes V0, . . . Vk−1 and let M be a matching in H. Let vi(E) = E ∩ Vi for an edge E ∈ H and for notational convenience all additions are in Z \ kZ. Let Ti = (vi, vi+1, . . . , vi+r−1) with i ∈ Z \ kZ and v_j ∈ V_j for all j ∈ {i, . . . , i + r − 1} and let E = (E₀, E1, . . . , Ek−r−1) ∈ [M ]k−r be a (k − r)-tuple of matching edges. We say Ti is adjacent to E (and vice versa) if

{vi, . . . , vi+r−1, vi+r(E0), . . . vi+k−1(Ek−r−1)} ∈ H. The set

N (Ti, (E0, . . . , Ek−r−1)) = {vi+r(E0), . . . vi+k−1(Ek−r−1)}

is called the neighbour of T with respect to E and by deg(Ti, [M ]k−r) we denote the number of (k − r)-tuples E ∈ [M ]_k−r the tuple Ti is adjacent to.

Proof of Theorem 6. For the proof keep in mind that all additions are considered in

N \ kN. Take M to be a largest matching in H. By adding arbitrary k-tuples if nec- essary, we may assume without loss of generality that |M | = n − r. Then there are

rk unmatched vertices which we divide into k pairwise disjoint sets T0, . . . , Tk−1 with

For an arbitrary edge E ∈ H we say E is M -non-crossing if there is an F ∈ M such that |E ∩ F | ≥ 2. Note that the number of M -non-crossing edges containing a fixed Ti is at most k|M |k−r−1. Hence, the restriction on the minimum r-degree implies

deg(Ti, [M ]k−r) ≥ δr(H) − knk−r−1>

k − r k n

k−r_.

And since this is true for each T_i, i ∈ {0, . . . , k − 1} the total degree is

deg(T0. . . Tk−1, [M ]k−r) :=

X

i∈{0,...,k−1}

deg(Ti, [M ]k−r) > (k − r)nk−r.

Then, by averaging, we conclude that there must be a (k − r)-tuple of matching edges (E₀, . . . , Ek−r−1) which is adjacent to at least (k − r + 1) tuples Ti. Without loss of generality let those Ti be T0, . . . , Tk−r. It is immediate from the definition that

N (Ti, (E0, . . . , Ek−r−1)) = {vi+r(E0), . . . vi+k−1(Ek−r−1)}, the neighbours of those Ti with respect to (E₀, . . . , Ek−r), are pairwise disjoint. And since each pair Ti and

N (Ti, (E0, . . . , Ek−r−1)) form an edge in H the (k − r + 1) tuples Ti and their neigh- bours N (Ti, (E0, . . . , Ek−r−1)) form a matching of size (k − r + 1) in H. Replacing

E0, . . . , Ek−r−1 by this matching we obtain a larger matching.

Proof of Theorem 6. Let n0 be as asserted by Lemma 42 for given k and r. Next let H

be a k-uniform hypergraph on n > n_{0 vertices, n ∈ kN, with minimum r-degree}

δr(H) ≥ k − r k n k − r ! + kk+1(ln n)1/2nk−r−1/2.

According to Lemma 42 there is a partition of V = V (H) into k partition classes

V = V0∪ . . . ˙˙ ∪Vk−1 such that |Vi| = |Vj| = n/k =: m for all i, j and every crossing

r-set T satisfies

deg0(T ) ≥ (k − r)!

kk−r δr(H) − 2(k ln n)

1/2_nk−r−1/2_.

Using (m)k−r≥ mk−r− mk−r−1Pi∈[k−r]i a simple calculation yields

deg0(T ) ≥ k − r

k m

k−r_{+ km}k−r−1

for all crossing r-sets T . By Theorem 47 this ensures a matching covering all but (r − 1)k vertices.

Proof of Theorem 7. Let γ > 0 and integers k > r > 0 be given. Applying Corollary 46

with γ₁ = γ/4 and k, r we obtain n0₀. Applying Theorem 6 with the same k and r we obtain n00₀. Set n₀ = max{n0₀, 2n00₀, 4k4k/γ2} and let H be a k-uniform hypergraph on