While doing research I came unto the following problem:
Given a hypergraph $H$ which is $r$-partite, $r$-uniform (each edge contains exactly $r$ vertices), $k$-regular (each vertex is contained in exactly $k$ edges) and $n$-balanced (each partition contains $n$ vertices).
Does H contain a perfect matching (an independent set of edges that covers all vertices)?
In literature I've found results by Aharoni, Haxell, Alon, Rödl and others, with sufficient conditions, but none seem to contain these hypothesis over the graph. Any suggestions or pointers to literature would be greatly appreciated.
Instead of asking wether H contains a perfect matching, what would be sufficient conditions for H to have a perfect matching? More specifically, what invariants are important in this kind of problem?
So far I've seen $\delta(H)$ and $|H|$ or $|V|$. I wonder if there are sufficient conditions which do not contain hypothesis over the size of partitions or of the graph.