**MSC:**- 05C70 Factorization, matching, covering and packing
- 05B40 Packing and covering, See also {11H31, 52C15, 52C17}

graphs such that each edge of $K_n$ occurs in exactly two of the

graphs and two graphs have precisely one edge in common.

ODCs of $K_n$ and their generalizations have been extensively

studied by several authors (e.g. \cite{ahl92,gmr97,le00}).

In this paper, we investigate ODCs where the graph to be

covered twice is $K_{n,n}$ and all graphs in the collection

are isomorphic to a given small graph $G$.

We prove that there exists an ODC of $K_{n,n}$ by all proper

subgraphs $G$ of $K_{n,n}$ for $1 \leq n \leq 9$, with two

genuine exceptions.