Coverings of the complete (di-)graph with n vertices by complete bipartite (di-)graphs with n vertices I

Let n be an integer, n >= 2. A set M"n of complete bipartite (di-)graphs with n vertices is called a critical covering of the complete (di-)graph with n vertices if and only if the complete (di-)graph is covered by the (di-)graphs of M"n, but of no proper subset of M"n. All possible cardinalities of critical coverings M"n are determined for all integers n and for undirected as well as directed graphs.