On Mutually Orthogonal Graph-Path Squares

A decomposition { } s G G G − = 0 1 1 , , ,   of a graph H is a partition of the edge set of H into edgedisjoint subgraphs s G G G − 0 1 1 , , ,  . If i G G ≅ for all { } i s ∈ − 0,1, , 1  , then  is a decomposition of H by G. Two decompositions { } n G G G − = 0 1 1 , , ,   and { } n F F F − = 0 1 1 , , ,   of the complete bipartite graph n n K , are orthogonal if, ( ) ( ) i j E G F = 1  for all { } i j n ∈ − , 0,1, , 1  . A set of decompositions { } k− 0 1 1 , , ,     of n n K , is a set of k mutually orthogonal graph squares (MOGS) if i  and j  are orthogonal for all { } i j k ∈ − , 0,1, , 1  and i j ≠ . For any bipartite graph G with n edges, ( ) N n G , denotes the maximum number k in a largest possible set { } k− 0 1 1 , , ,     of MOGS of n n K , by GG. Our objective in this paper is to compute ( ) N n G , where ( ) d G F + = 1  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).


Introduction
In this paper we make use of the usual notation: for the union of D and F with v L (set of vertices) that belong to each other (i.e. union of D and F with common vertices of the set v L belong to F and D), n K for the complete graph on n vertices, 1 K for an isolated vertex. The other terminologies not defined here can be found in [1].
A decomposition of a graph H is a partition of the edge set of H into edge-disjoint sub-  [7] have proved El-Shanawany conjectured. In the following section, we present another technique to prove this conjecture as in Theorem 8.
The two sets x y of , n n K is defined to be the difference y x − , where , n x y ∈  . Note that sums and differences are carried over in n  (that is, sums and differences are carried modulo n). Let G be a subgraph of , n n K without isolated vertices and let is half-starter if ( ) E G n = and the lengths of all edges in G are mutually different. Lemma 2 (see [8]). If G is a half-starter, then the union of all translates of G forms an edge decomposition of In what follows, we denote a half-starter G by the vector ( ) ( ) Theorem 3 (see [8]). Two half-starters ( ) ( ) In the following, we define a G-square over additive group n  . Definition 4 (see [6]). Let G be a subgraph of , n n K A square matrix  of order n is called an G-square if every element in n  occur exactly n times, and the graphs i G , We have already from Lemma 2 and Definition 4 that every half starter vector ( ) v G and its translates are equivalent to G-square. For more illustration, the first matrix 0  in equation (1) is equivalent to the first row in . Now, we shall derive a class of mutually orthogonal subgraphs of , n n K by a given graph G as follow.
is called a set of k mutually orthogonal graph squares (MOGS) if i  and j  are orthogonal for all and i j ≠ . Definition 7 (see [9]). Let F be a certain graph, the graph F-path denoted by , is a path of a set of vertices and a set of edges if and only if there exists the following two bijective mappings: is a collection of d graphs, each one is isomorphic to the graph F.
2) : is a class of disjoint sets of vertices (i.e.,  decomposed into 1 d + disjoint sets such that no two vertices within the same set are adjacent).
As a special case if the given graph F is isomorphic to 1,1 K then ( ) , is the natural path   ( ) 6 1,3 K  , the path of 6 sets of vertices and 5 edges of 1,3 K .

Mutually Orthogonal Graph-Path Squares
The following result was shown in [7]. Here we present another technique for the proof. , .
with q edges; for fixed q j ∈  and 0 1 i q ≤ ≤ − , define the q halfstarter vectors as follows, ( ) our task is to prove the orthogonality of those q half-stater vectors in mutually. Let us define the half starter vector ( ) Then for all two different elements , It remains to prove the isomorphism of ( ) , and therefore k l = . Moreover, for any , q i j ∈  the th ij graph isomorphic to ij G has the edges: − + ■ An immediate consequence of the Theorem 8 and Conjecture 1 is the following result. Example 9. The three mutually orthogonal decompositions (MOD) of 3,3 K by 4 P given in Figure 3 are associated with the three mutually orthogonal 4 P -squares as in Equation (1) Note that, every row in Figure 3 represents edge decompositions of 3,3 K by 4 P .
Our task is to prove the orthogonality of those q half-starter vectors in mutually. Let us define the half starter Then for all two different elements , , and therefore k l = . Moreover, for any 0 1, n i q j ≤ ≤ − ∈ the th ij graph ij G isomorphic to G has the edges: 3,3 if q is a prime number with an integer power 1 x ≥ and , s k are positive integers.