Topology of complexes of edge covering partite graphs and hypergraphs (Q1715078)

The vertex set \(A\) of an \(m\)-partite \(r\)-hypergraph of type \((k_1,\ldots,k_m)\) partitions into \(m\) subsets \(A_i\) of orders \(k_1,\ldots,k_m\) and the edges are the \(r\)-subsets of \(A\) all \(r\) elements of any of which belong to different sets \(A_i\). An \(m\)-partite \(r\)-hypergraph is edge-covering if each vertex of the corresponding \(m\)-partite graph belongs to at least one hyperedge. Each \(m\)-partite \(r\)-hypergraph corresponds to a face of the simplex whose vertices are all the \(r\)-element hyperedges on \(A\); the face is spanned by its hyperedges. The faces spanned by the non-edge-covering \(m\)-partite \(r\)-hypergraphs form the simplicial subcomplex \( {\mathcal{H}}_r(A) \). The aim of this article is the calculation of the homology groups and homotopy types of the two related complexes formed by the edge-covering and non-edge-covering \(m\)-partite \(r\)-hypergraphs.\par In the main result of the article, it is shown that \( {\mathcal{H}}_r(A) \) is homotopy equivalent to the wedge of \( \binom{m-1}{r-1} \) spheres of dimension \(\vert A\vert -r-1\), it is acyclic in all positive dimensions except for \(\vert A\vert -r-1\), its \((\vert A\vert -r-1)\)-homology group is \( {\mathbb{Z}}^\binom{m-1}{r-1} \), and its Euler characteristic is \( 1 - (-1)^{\vert A\vert -r}\binom{m-1}{r-1} \). The factorization of the entire simplex by \( {\mathcal{H}}_r(A) \) is a cell complex whose cells correspond to the edge-covering \(r\)-hypergraphs and the \(i\)-th homology group of this complex is trivial except for the case \(i=\vert A\vert -r\) when it is the group \( {\mathbb{Z}}^\binom{m-1}{r-1} \). The factoring's Euler characteristic is equal to \( 1 + (-1)^{\vert A\vert -r}\binom{m-1}{r-1} \). Beside providing the proofs and some further results, the author considers in more detail the two special cases \( m=r \) and \( \vert A\vert =m \).