**MSC:**- 05C70 Factorization, matching, covering and packing
- 05B30 Other designs, configurations, See also {51E30}

said to be a suborthogonal double cover (SODC) of the complete graph

$K_n$ by $G$, if every edge of $K_n$ belongs to exactly two members

of $\mathcal{G}$ and any two different elements from $\mathcal{G}$

share at most one edge. We introduce the notion of an SODC as a

generalization of the well-known concept of orthogonal double covers.

Our objective is to investigate the sets $S(G)$ of integers $n$

allowing an SODC of $K_n$ by a given graph $G$. We shall prove that

$S(G)$ contains almost all integers $n$ satisfying certain necessary

conditions.