On 2-perfect cube decompositions

A cube decomposition Q of a graph G is said to be 2-perfect if for every edge {x, y} ∈ E(G), x and y are connected by a path of length 1 in exactly one cube of Q, and are also connected by a path of length 2 in exactly one (distinct) cube of Q. For both Kv and Kv −F , we give constructions for half of the cases which satisfy the obvious necessary conditions, with a small number of exceptions. We also show that there does not exist a 2-perfect Q-decomposition of K16 and conjecture that the same is true for K25.