This is in relation to the question here. What, if any, are the known conditions for the existence of a perfect matching for a $r$-regular, $r$-uniform, $r$-partite hypergraph. I specifically interested in the $r=3$ case, but any general information is good. So far I have not been able to find a reference that discusses this particular case.

In the question link to above, he is not requiring that the $k$ (for $k$-regular) be the same as the $r$ (for $r$-uniform and $r$-partite), and the example given in the solution uses a case where $k\neq r$.

Any indication on where I might find references for this case or even just the theorems themselves (with the references) would be greatly appreciated.