In this paper, we give an introduction to an area of combinatorial theory which deals with. mutually orthogonal  (MOGSs) where  is a spanning subgraph of  That is, we will investigate the maximal number of a collection of  of order n  which are orthogonal in pairs. The most general object studied in this theory is a set of the so-called mutually orthogonal Latin squares (MOLSs) of order  which are equivalent to mutually orthogonal if . That is, this approach is an extension of MOLSs. Our objective in this paper is to discuss two direct constructions from conjecture in [7] for all prime numbers   to obtain MOGSs  where .