Orthogonal Double Covers of Complete Bipartite Graphs by Certain Copies of Cyclic and one Factorization Graphs

Let K
_n,n
be a complete bipartite graph on 2n vertices and G be a collection of 2n subgraphs of K
_n,n
, then G is an orthogonal double cover (ODC) of K
_n,n
if every edge of K
_n,n
exists in exactly two members of G and any two members of G share exactly an edge. If all subgraphs in G are isomorphic to a given subgraph G, then G is said to be an ODC of K
_n,n
by G. Our aim is to get an ODC of K
_n,n
by ∪Cs; s ≤ n, and we will make extension for this ODC to get another one of K
_m,m
by ∪(αCs); where m = αn; α ∈ Z
⁺
.