Collapsibility of simplicial complexes of hypergraphs

Let $\mathcal{H}$ be a hypergraph of rank $r$. We show that the simplicial complex whose simplices are the hypergraphs $\mathcal{F}\subset\mathcal{H}$ with covering number at most $p$ is $\left(\binom{r+p}{r}-1\right)$-collapsible, and the simplicial complex whose simplices are the pairwise intersecting hypergraphs $\mathcal{F}\subset\mathcal{H}$ is $\frac{1}{2}\binom{2r}{r}$-collapsible.


Introduction
Let X be a finite simplicial complex. Let σ ∈ X such that |σ| ≤ d and σ is contained in a unique maximal face τ ∈ X. We say that the complex X ′ = X \ {η ∈ X : σ ⊂ η ⊂ τ } is obtained from X by an elementary d-collapse, and we write The complex X is d-collapsible if there is a sequence of elementary dcollapses from X to the void complex ∅. The sequence is called a d-collapsing sequence for X. The collapsibility of X is the minimal d such that X is d-collapsible.
A simple consequence of d-collapsibility is the following (see [9]). Claim 1.1. If X is d-collapsible then X collapses to a complex of dimension smaller than d. In particular, the homology groupsH k (X) are trivial for k ≥ d.
Let H be a finite hypergraph. We identify H with its edge set. The rank of H is the maximal size of an edge of H.
A set C is a cover of H if A ∩ C = ∅ for all A ∈ H. The covering number of H, denoted by τ (H), is the minimal size of a cover of H. So Cov H,p is a simplicial complex whose vertices are the edges of H and whose simplices are the hypergraphs F ⊂ H that can be covered by a set of size at most p. Some topological properties of the complex Cov ( [n] r ),p were studied by Jonsson in [5]. The So Int H is a simplicial complex whose vertices are the edges of H and whose simplices are the hypergraphs F ⊂ H that are pairwise intersecting.
Our main results are the following: The proofs rely on two main ingredients. The first one is a general construction of a d-collapsing sequence for a simplicial complex (with d depending on the complex), due essentially to Matoušek and Tancer (who stated it in the special case where the complex is the nerve of a family of finite sets, and used it to prove the case p = 1 of Theorem 1.2).
The second ingredient is the following combinatorial lemma, proved independently by Frankl and Kalai. Lemma 1.4 (Frankl [3], Kalai [6]). Let {A 1 , . . . , A k } and {B 1 , . . . , B k } be families of sets such that: The paper is organized as follows. In Section 2 we present the dcollapsing sequence of Matoušek and Tancer. In Section 3 we present some results on the collapsibility of independence complexes of graphs. In Section 4 we prove our main results on the collapsibility of complexes of hypergraphs. In Section 5 we present some generalizations of Theorems 1.2 and 1.3, that are obtained by applying different known variants of Lemma 1.4.

A d-collapsing sequence for a simplicial complex
Let X be a simplicial complex on vertex set V . Fix an arbitrary linear order < on the vertices V . Let σ 1 , . . . , σ m be the maximal faces of X. For a simplex σ ∈ X let m(σ) = min{i ∈ [m] : σ ⊂ σ i }.
Let i ∈ [m] and σ ∈ X such that m(σ) = i. We define the minimal exclusion sequence mes(σ) = (v 1 , . . . , v i−1 ) as follows: If i = 1 then mes(σ) is the empty sequence. If i > 1 we define the sequence recursively as follows: Since i > 1, we must have σ ⊂ σ 1 , hence there is some v ∈ σ such that v / ∈ σ 1 . Let v 1 be the minimal such vertex (with respect to the order <). Let j < i and assume that we already defined v 1 , . . . , v j−1 . Since i > j, we must have σ ⊂ σ j , hence there is some v ∈ σ such that v / ∈ σ j .
• If there is such a vertex v k ∈ {v 1 , . . . , v j−1 }, let v j be such a vertex with minimal k. In this case we call v j old at j.
• If v k ∈ σ j for all k < j, then let v j be the minimal vertex v ∈ σ (with respect to the order <) such that v / ∈ σ j . In this case we call v j new at j.
Let M (σ) be the simplex consisting of all the vertices appearing in the sequence mes(σ). Note that mes(M (σ)) = mes(σ). Let The following result was stated and proved in [8,Prop. 1.3] in the special case where X is the nerve of a finite family of sets (in our notation, X = Cov H,1 for some hypergraph H), but the proof given there can be easily modified to hold in a more general setting.
For completeness we include here the proof.
We define a linear order ≺ on M as follows: First, we order the families M i by decreasing i: The simplices in M m come first, then the ones in M m−1 and so on. Within each M i we order the simplices lexicographically by their minimal exclusion sequence.

Let
where η ′ is the element of M succeeding η in the order ≺.

By Lemma 2.4 we have
where η ′ is the element of M succeeding η in the order ≺.
Note that Y 1 = X. Recall that d(X) = max i∈[s] |η i |. By Lemma 2.5 we have the following d(X)-collapsing sequence: Thus X is d(X)-collapsible.

Collapsibility of independence complexes
Let G = (V, E) be a graph. The independence complex I(G) is the simplicial complex on vertex set V whose simplices are the independent sets in G.
Definition 3.1. Let k(G) be the maximal size of a set {v 1 , . . . , v k } ⊂ V that satisfies: • There exist u 1 , . . . , u k ∈ V such that Proof. Let X = I(G), and let σ 1 , . . . , σ m be the maximal faces of X (i.e. the maximal independent sets of G). Let i ∈ [m] and σ ∈ X with m(σ) = i, that is σ ⊂ σ i and σ ⊂ σ j for j < i.
As a simple corollary we obtain Proof. Let k = k(G) and let v 1 , . . . , v k , u 1 , . . . , u k ∈ V satisfying the conditions in Definition 3.1. Then the vertices v 1 , . . . , v k , u 1 , . . . , u k must be all distinct, therefore 2k ≤ n. Thus so the claim follows from Proposition 3.2.

Complexes of hypergraphs
Next we prove our main results, Theorems 1.2 and 1.3.

Proof of Theorem 1.2. Let H be a hypergraph of rank r on vertex set [n]
and let X = Cov H,p . Let By the definition of the minimal exclusion sequence we have Also, for all j ∈ [k] we have A i j ∩ C i = ∅, since F ∈ X i . Therefore the pair of families Let k = k(G) and let {A 1 , . . . , A k } ⊂ H that satisfies the conditions of Definition 3.1. That is, • There exist B 1 , . . . , B k ∈ H such that Then the pair of families We obtain the following: The complex Cov t+1 H 1 is the boundary of the r+p−2t r−t − 2 -collapsible, and the complex Int t+1 H 2 is the boundary of the 1 2 2(r−t) r−t -dimensional cross-polytope, hence it is not r−t − 1 -collapsible. Restricting ourselves to special classes of hypergraphs we may obtain better bounds on the collapsibility of their associated complexes. For example, we may look at r-partite r-uniform hypergraphs (that is, hypergraphs H on vertex set V = V 1 · ∪ V 2 · ∪ · · · · ∪ V r such that |A ∩ V i | = 1 for all A ∈ H and i ∈ [r]). In this case we have the following result: Theorem 5.4. Let H be an r-partite r-uniform hypergraph. Then Int H is 2 r−1 -collapsible.
The next example shows that the bound on the collapsibility of Int H in Theorem 5.4 is tight: Let H be the complete r-partite r-uniform hypergraph with all sides of size 2. It has 2 r edges, and any edge A ∈ H intersects all the edges of H except its complement. Therefore the complex Int H is the boundary of the 2 r−1 -dimensional cross-polytope, so it is homeomorphic to a (2 r−1 − 1)-dimensional sphere. Hence, by Claim 1.1, Int H is not (2 r−1 − 1)collapsible.
For the proof we need the following Lemma, due to Lovász, Nešetřil and Pultr. ∪ V 2 · ∪ · · · · ∪ V r such that: • A i ∩ B j = ∅ for all 1 ≤ i < j ≤ k.
A common generalization of Lemma 1.4 and Lemma 5.5 was proved by Alon in [2].
The proof of Theorem 5.4 is the same as the proof of Theorem 1.3, except that we replace Lemma 1.4 by Lemma 5.5. A similar argument was also used by Aharoni and Berger ([1, Theorem 5.1]) in order to prove a related result about rainbow matchings in r-partite r-uniform hypergraphs.