On Construction Mutually Orthogonal Disjoint Unions of Small Certain Trees Graph Squares

A family of decompositions $\mathcal{G} = \left\{ {{\mathcal{G}_0},{\mathcal{G}_1}, \ldots \ldots ,{\mathcal{G}_{{\text{r}} - {\text{l}}}}} \right.\} $ of a complete bipartite graph K
n
, n is a set of r mutually orthogonal graph squares (MOGS) if ${\mathcal{G}_i}$ and ${\mathcal{G}_j}$ are orthogonal for all i j, ∈{0, 1, … −, r 1}, and i ≠ j. For any subgraph G of K
n
n, with n edges, N (n G, ) denotes the maximum number r in a largest possible set $\mathcal{G} = \left\{ {{\mathcal{G}_0},{\mathcal{G}_1}, \ldots \ldots ,{\mathcal{G}_{{\text{r}} - {\text{l}}}}} \right.\} $ of MOGS of K
n
, n by G . In this paper, we compute two extensions N (n, G) = r ≥ 4, for n = 11, we have G = (4K
1,2
∪3K
2
), and n = 13, G will be (3K
1,2
∪7K
2
).