On Mutually Orthogonal Disjoint Copies of Graph Squares

Abstract

A family of decompositions $\{\mathcal{G}_{0},\mathcal{G}_{1},...,% \mathcal{G}_{k-1}\}$ of a complete bipartite graph $K_{n,n}$ is a set of $k$ \textit{mutually orthogonal graph squares} $(MOGS)$ if $\mathcal{G}_{i}$ and $\mathcal{G}_{j}$ \ are orthogonal for all $i,j\in \{0,1,...,k-1\}$ and $% i\neq j$. For any subgraph $G$ of $K_{n,n}$ with $n$ edges, $N(n,G)$ denotes the maximum number $k$ in a largest possible set $\{\mathcal{G}_{0},\mathcal{%G}_{1},...,\mathcal{G}_{k-1}\}$ of $(MOGS)$ of $K_{n,n}$ by $G$. Our objective of this paper is to compute $N(n,G)=k\geq 3$ where $G$ represents disjoint copies of certain subgraphs of $K_{n,n}$.