The existence of room 5-cubes

A Room t-cube is standardized if it also satisfies (iv) the ith diagonal cell contains the pair of symbols {co, i}. There are several other equivalent forms that are taken by Room t-cubes, one in terms of graphs and one in terms of Latin squares. The first of these forms is as a set of orthogonal one-factorizations of K,,. Let G be a graph with an even number of vertices. A one-factor in G is a set of (pairwise disjoint) edges which between them contain each vertex exactly once. A one-factorization is a partition of all the edges of G into pairwise disjoint one-factors. Two one-factorizations F, and F2 are orthogonal if any one-factor in F, and any one-factor in F2 have at most one edge in common. The following theorem from [12] shows the connection between orthogonal one-factorizations of K, and Room t-cubes.