Orthogonally Resolvable Matching Designs

An orthogonally resolvable matching design OMD(n,k) is a partition of the edges of the complete graph Kn into matchings of size k, called blocks, such that the blocks can be resolved in two different ways. Such a design can be represented as a square array whose cells are either empty or contain a matching of size k, where every vertex appears exactly once in each row and column. In this paper we show that an OMD(n,k) exists if and only if n≡0(mod2k) except when k=1 and n=4 or 6.