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A decomposition of a graph H is a partition of the edge set of H into edge-disjoint subgraphs . If for all , then **G** is a decomposition of H by G. Two decompositions and of the complete bipartite graph are orthogonal if, for all . A set of decompositions of is a set of k mutually orthogonal graph squares (MOGS) if and are orthogonal for all and . For any bipartite graph G with n edges, denotes the maximum number k in a largest possible set of MOGS of by G. Our objective in this paper is to compute where is a path of length d with d + 1 vertices (i.e. Every edge of this path is one-to-one corresponding to an isomorphic to a certain graph F).

In this paper we make use of the usual notation:

A decomposition

thogonal if,

for all

If two decompositions

It is well-known that orthogonal Latin squares exist for every

El-Shanawany [

number, then

Conjecturer 1. Let p be a prime number. Then

Sampathkumar et al. [

The two sets

length of the edge

if

Lemma 2 (see [

In what follows, we denote a half-starter G by the vector

where

Theorem 3 (see [

If two half-starters

A set of decompositions

thogonal graph squares (MOGS) if

Note that

In the following, we define a G-square over additive group

Definition 4 (see [

We have already from Lemma 2 and Definition 4 that every half starter vector

Definition 5. Two squares matrices

Now, we shall derive a class of mutually orthogonal subgraphs of

Definition 6. A set of matrices

Definition 7 (see [

1)

2)

As a special case if the given graph F is isomorphic to

For more illustration, see

Consider

In the following section, we will compute

The following result was shown in [

Theorem 8. Let q be a prime number. Then

Proof. Let

vectors of graphs

of

An immediate consequence of the Theorem 8 and Conjecture 1 is the following result.

Example 9. The three mutually orthogonal decompositions (MOD) of

Note that, every row in

The following result is a generalization of the Theorem 8.

Theorem 10. Let n be a prime power such that

Proof. For fixed

Our task is to prove the orthogonality of those q half-starter vectors in mutually. Let us define the half starter vector

have

half-stater vectors of graphs

of

■

Note that, in the special case

Furthermore, we can construct the following result using Theorem 10 in case

Theorem 11. Let

of

Proof. The result follows from the vector in Equation (2) and its edges in Equation (3) with

number of graphs isomorphic to F. As a direct application of Theorem 11; see

Conjecture 12.

Conjecture 13.

RamadanEl-Shanawany, (2016) On Mutually Orthogonal Graph-Path Squares. Open Journal of Discrete Mathematics,06,7-12. doi: 10.4236/ojdm.2016.61002