Construction of Mutually Orthogonal Graph Squares Using Novel Product Techniques

Sets of mutually orthogonal Latin squares prescribe the order in which to apply diferent treatments in designing an experiment to permit efective statistical analysis of results, they encode the incidence structure of fnite geometries, they encapsulate the structure of fnite groups and more general algebraic objects known as quasigroups, and they produce optimal density error-correcting codes. Tis paper gives some new results on mutually orthogonal graph squares. Mutually orthogonal graph squares generalize orthogonal Latin squares interestingly. Mutually orthogonal graph squares are an area of combinatorial design theory that has many applications in optical communications, wireless communications, cryptography, storage system design, algorithm design and analysis, and communication protocols, to mention just a few areas. In this paper, novel product techniques of mutually orthogonal graph squares are considered. Proposed product techniques are the half-starters’ vectors Cartesian product, half-starters’ function product, and tensor product of graphs. It is shown that by taking mutually orthogonal subgraphs of complete bipartite graphs, one can obtain enough mutually orthogonal subgraphs in some larger complete bipartite graphs. Also, we try to fnd the minimum number of mutually orthogonal subgraphs for certain graphs based on the proposed product techniques. As a direct application to the proposed diferent product techniques, mutually orthogonal graph squares for disjoint unions of stars are constructed. All the constructed results in this paper can be used to generate new graph-orthogonal arrays and new authentication codes.


Introduction
Graphs are discrete structures consisting of vertices and edges that connect these vertices.Several problems in almost every conceivable discipline can be solved using graph models.Certain problems in physics, chemistry, computer technology, psychology, communication science, linguistics, engineering, sociology, and genetics can be formulated as problems in graph theory.For instance, graphs are used to represent the competition of diferent species in an environment, to represent who infuences whom in an organization, and to represent the outcomes of roundrobin tournaments.Also, graphs are used to model relationships between people, collaborations between researchers, telephone calls between telephone numbers, and links between websites, to mention just a few areas.Many branches of mathematics, such as probability, topology, matrix theory, and group theory, have strong connections with graph theory.For standard terminology and notations concerning graph theory, see [1].Decompositions of complete bipartite graphs have several applications in the design of experiments, graph code generation, and authentication codes [2,3].Table 1 shows the nomenclature used in the paper.
In this paper, we are concerned with an area of combinatorial theory that deals with mutually orthogonal F squares where F is a subgraph of K n,n .Mutually orthogonal Latin squares (MOLS) are a special case of mutually orthogonal graph squares (MOGS).MOGS are interesting but not attainable for general graphs.Combinatorial design theory has many applications in optical communications, wireless communications, cryptography, storage system design, algorithm design and analysis, and communication protocols, to mention just a few areas.
Defnition 1 (see [4]).Let F be a subgraph of K n,n with size n.A square matrix M of order n is an F− square if every element in Z n � 0, 1, . . ., n − 1 { } is found exactly n times in M, and the graphs F k with E(F k ) � (a 0 , b 1 ): M(a 0 , b 1 ) �  k: k ∈ Z n } are isomorphic to F. Te elements of Z n × 0 { } are used for labeling the rows of M, and the elements of Z n × 1 { } are used for labeling the columns of M.An edge decomposition of K n,n by a graph F can be represented by an F− square.
Defnition 2 (see [4]).Suppose M 1 is an F− square of order n with entries from a set C, and M 2 is an F− square of order n with entries from a set D. Ten, the two squares M 1 and M 2 are orthogonal if, for every x ∈ C and for every y ∈ D, there exists exactly one cell (a 0 , b 1 ) such that M 1 (a 0 , b 1 ) � x and M 2 (a 0 , b 1 ) � y.A set of λF− squares of order n, say M 1 , . . ., M λ , are called pairwise orthogonal (mutually orthogonal) F− squares (MOGS) if M p and M q are orthogonal for all 1 ≤ p < q ≤ λ.Here, we consider C � D � Z n � 0, 1, . . ., n − 1 { }.
Theorem 1.For the bipartite graph F having n edges, N(n, F) denotes the maximum number k in a largest possible set of MOGS of K n,n by F. For every bipartite graph F with n ≥ 2 edges, we have N(n, F) ≤ n.
Great eforts have been made to get the solution to several problems concerned with the MOLS since Euler frst asked about MOLS to solve the thirty-six ofcer's problem.Famous theorems concerning the MOLS were introduced by Bose, Shrikhande, and Parker [5,6].Also, Wilson in [7] handled celebrated theorems concerned with the MOLS.Many eforts have been concentrated on refning and fnding novel applications for these approaches.Te authors of [8] proposed an integrated frefy algorithm based on MOLS, named FA-MOLS, to address the quadratic assignment problem.Liu [9] introduced the packing of Latin squares by BCL algebras.Te authors in [10] focused on the existence of orthogonal large sets of partitioned incomplete Latin squares.A large set of disjoint incomplete Latin squares was introduced in [11].A strategy for producing group-based Sudoku-pair Latin squares was investigated in [12].Te Latin squares were constructed based on the circulant matrix by the authors of [13].Authentication codes based on orthogonal arrays and Latin squares were proposed in [14].For a good survey of MOLS, see [15] and the references therein.El-Shanawany [16] proposed the conjecture, N(p, P p+1 ) � p, where P p+1 is a path with p + 1 vertices and p is a prime number.Sampathkumar et al. [17] solved this conjecture.El-Shanawany [18] found N(p, P p+1 (F)).El-Shanawany [19] computed N(n, F) � r ≥ 3 where F is disjoint copies of some subgraphs of K n,n .El-Shanawany and El-Mesady [4] introduced the Kronecker product of MOGS and applied this technique to get a new mutually orthogonal disjoint union of some complete bipartite graph squares.MOGS for disjoint unions of paths were developed in [20].MOGS for certain graphs were handled by [21].El-Mesady et al. [22] generalized the MacNeish's Kronecker product theorem of MOLS.MOGS were used to construct graph-transversal designs and graphauthentication codes in [3,23].MOGS are used to construct orthogonal arrays that have many applications [24].
Te main purpose of this paper is to construct several new results on MOGS.All of the previously mentioned MOGS results motivated us to introduce novel diferent product techniques to MOGS that yield new MOGS results.Te proposed product techniques are the half-starters' vectors Cartesian product, half-starters' function product, and graph tensor product.Te novelty of the current paper is demonstrated by the fact that it is the frst to introduce the MOGS by the aforementioned product techniques.It is shown that by taking mutually orthogonal subgraphs of complete bipartite graphs, one can obtain enough mutually orthogonal subgraphs in some larger complete bipartite graphs.Also, we try to fnd the minimum number of mutually orthogonal subgraphs for certain graphs based on the proposed product techniques.As a direct application to the proposed diferent product techniques, mutually orthogonal graph squares for disjoint unions of stars are constructed.Te main diference between this paper and almost all the related study works that we surveyed in this section is that the proposed product techniques are recursive construction techniques that can use all the results in the literature to construct novel results concerned with MOGS.Also, the Kronecker product [4] was applied to the squares, but the half-starters' vectors Cartesian product is applied to the vectors, the halfstarters' function product is applied to the functions, and the graph tensor product is applied to graphs.
Te remaining part of the present paper is divided as follows: Section 2 is devoted to MOGS from mutually orthogonal halfstarters' vectors.Section 3 constructs MOGS based on the Cartesian product of half-starters' vectors.MOGS from mutually orthogonal half-starters' functions are presented in Section 4. Section 5 introduces the tensor products of MOGS.MOGS for complete bipartite graphs by stars based on the tensor product are proved in Section 6. Discussion is presented in Section 7. Section 8 is devoted to the conclusion and future work.

MOGS from Mutually Orthogonal Half Starters' Vectors
If we have a graph F which is considered a subgraph of K n,n with n edges, then the graph Theorem 2 (see [9]).If F is a half-starter, then an edge decomposition of K n,n can be constructed by fnding all the translates of F and taking their union; that is, √√√√√√√√√√ n− times can be used to represent the half-starter F where u k , k ∈ Z n , and (u k ) 0 is the unique vertex and u(F l ) are orthogonal for every 0 ≤ k < l ≤ λ − 1.It is worth noting that each half-starter and its translates of a subgraph F of K n,n are equivalent to F− square.Hence, the set of k mutually orthogonal half-starters and their translates are equivalent to a set of k mutually orthogonal F− squares.

MOGS Based on the Cartesian Product of Half-Starters' Vectors
Te Cartesian product of half-starters' vectors has been defned in literature for constructing orthogonal double covers of K n,n .Tis method has been applied to construct orthogonal double covers of K n,n by new graph classes.Te Cartesian product of two vectors corresponding to two halfstarter graphs is considered a very special case of the tensor product of these two half-starter graphs.
Defnition 3. Te tensor product of two graphs G 1 and Example 1. Figure 1 exhibits an example of the graphs Defnition 4 (see [25]).Let G be a graph, and v belongs to the vertex set of G. Te number of edges incident at v in G is called the degree (or valency) of the vertex v in G and is denoted by d(v).From the degrees of vertices of G, we can construct a sequence which is called a degree sequence of G, when the vertices are taken in the same order.It is customary to put this sequence in nondecreasing or nonincreasing order.Tis gives a unique sequence.
Defnition 5.If we have the vector u(G) � (u 0 , u 1 , . . ., u (m− 1) ) ∈ Z m m , then, by determining the repetition number of each element in the vector u(G), we get the vector N � ((n(0), n(1), . . ., n((m − 1))), where n(i) is the repetition number of the element i, i ∈ Z m .By the ascending order for the vector N, we get the degree sequence of the vector u(G) defned by L � (l 0 , l 1 , . . ., l (m− 1) ), where l 0 ≤ l 1 ≤ . . .≤ l (m− 1) .Defnition 6.If we have the two vectors u p (G p ) � (u then the two half-starters G p and G q are isomorphic if L p � L q and L P ′ � L q ′ or L p � L q ′ and L p ′ � L q , where L p is the degree sequence of the vector u p , L p ′ is the degree sequence of the vector u ′ p ,L q is the degree sequence of the vector u q , and L q ′ is the degree sequence of the vector u ′ q .In our paper, we consider the case of L p � L q and L P ′ � L q ′ for the isomorphism of the two half-starters G p and G q . For Proposition 1, if we have the two half-starters G and H which are represented by the vectors v(G) ∈ Z m m and u(H) ∈ Z n n , respectively, then the graph T� G ⊗ H is defned by the edge set

Journal of Mathematics
Hence, from (3), the two half-starters T p and T q are orthogonal.Since Now, we will try to prove the isomorphism of the graphs Since the degree sequence of the vector v p equals the degree sequence of the vector v q , the degree sequence of the vector v ′ P equals the degree sequence of the vector v ′ q , the degree sequence of the vector u p equals the degree sequence of the vector u q , and the degree sequence of the vector u ′ p equals the degree sequence of the vector u ′ q , then the degree sequence of the vector v p u p equals the degree sequence of the vector v q u q , and the degree sequence of the vector v ′ p u ′ p equals the degree sequence of the vector v ′ q u ′ q .Hence, the two halfstarters T p and T q are isomorphic.

MOGS from Mutually Orthogonal Half-Starters' Functions
In what follows then the graphs G f i will be represented by the functions Every graph from the graphs G f i represents unions of stars which have the same direction where every vertex x belongs to V 1 has a degree of one, that is, Defnition 7 (see [9]).Let z ∈ Z n .Ten, the graph Remark 1 (see [9]).Te union of all translates of G f forms an edge decomposition of K n,n , that is, Defnition 7 and Remark 1 show that every f− halfstarter graph G f and the translates are equivalent to G f -square. Let If the two half-starters G f and G g are orthogonal, then the two sets of translates of G f and G g are orthogonal.A set of edge decompositions Te edge set of the graphs G f i and their translates are shown in Tables 3, 4,  and 5.   Journal of Mathematics Hence, we deduce the following three mutually orthogonal (K 2 ∪ K 1,2 )-squares: By the ascending order for the vector N, we get the degree vector of G f denoted by L f � (l 0 , l 1 , . . ., l n− 1 ) where where L f is the degree vector of G f and L g is the degree vector of G g .
We shall denote by N(n, G f i ) the maximal number of f i − half-starter graphs G f i in the largest possible set of mutually orthogonal subgraphs . Also, we present some results as direct applications to Proposition 2. In the following, if there is no danger of ambiguity, if (x, y) ∈ Z m ⊗ Z n , we can write (x, y) as xy.
which are represented by the functions Since h p (xy) − h q (xy) � f p (x)  g p (y) − f q (x)g q (y) � (f p (x) − f q (x))(g p (y) − g q (y)): p, q ∈ Z k , p ≠ q} � Z m ⊗ Z n , then G hp and G hq are orthogonal.Te edge set of the graphs G h i can be obtained as follows, since .Now, we want to prove the isomorphism of the two graphs G hp and G hq .For p, q ∈ Z k , p ≠ q, we have L f p � L f q and L g p � L g q , then for G hp � G f p ⊗ G g p and (2, 2) (2, 0) 00 0 10 0 42 0 02 0 40 0 12 0 41 0 01 0 30 0 40 0 31 0 00 0 20 1 11 1 21 1 00 1 The graph corresponding to the position s having 00 values in L 0 .
The graph corresponding to the position s having 00 values in L 1 .
The graph corresponding to the position s having 00 values in L 2 .
Figure 3: Tree mutually orthogonal half-starters corresponding to the vectors w s (P s 6 ⊗ P s 4 ).

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G hq � G f q ⊗ G g q , the degree vector of G hp is equal to the degree vector of G hq .Tis means that L h p � L h q .Hence, G hp and G hq are isomorphic.
All the following results are based on (i) Proposition 2 and (ii) the following ingredients (see [9]).
(i) Let n > 2 be a prime number.Ten, N(n, K 1,1 ∪ ((n − 1)/2)K 1,2 ) � n, and the represented by the functions Tese known f i − half-starter graphs G f i are the ingredients for the following results.Tese ingredients are some of the literature results and are not all of the literature results.

Theorem . Let n, m be odd primes. Ten, N(mn, G
, which are represented by the functions f i (x) � x 2 + ix, where x ∈ Z m , i ∈ Z k (ingredient (i)) and k mutually orthogonal g i − half-starter graphs G g i � nK 1,1 , which are represented by the functions g i (y) � (i + 1)y, where y ∈ Z n ,i ∈ Z k (ingredient (ii)).Ten, we obtain k mutually orthogonal h i − half-starter graphs G h i � G f i ⊗ G g i , which are represented by the functions h i (xy) � (x 2 + ix)((i + 1)y).
Ten, we obtain k mutually orthogonal h i − half-starter graphs G h i � G f i ⊗ G g i , which are represented by the functions h i (xy) � (x 2 + ix)g i (y).Since h  p(xy) − h p (xy) � (x 2 + px)g p (y) − (x 2 + qx)g q (y) � ((x 2 + px)− (x 2 + qx))(g p (y) − g q (y)): p, q ∈ Z k , p ≠ q} � Z m ⊗ Z n , then G hp and G hq are orthogonal.Te edge set of the graphs G h i can be obtained as follows, since .Now, we want to prove the isomorphism of the two graphs G hp and G hq .For p, q ∈ Z k , p ≠ q, we have L f p � L f q � (0, 0, . . ., 0 g q , the degree vector of G hp is equal to the degree vector of G hq .Tis means that L f p � L f q � (0, 0, . . ., 0 , 1, 2, 2, . . ., 2) , 2).Hence, G hp and G hq are isomorphic.
Proof 7. We have ((m − 1)) mutually orthogonal f i − halfstarter graphs G f i � mK 1,1 , which are represented by the functions f i (x) � (i + 1)x, where x ∈ Z m , i ∈ Z k (ingredient (ii)) and (n − 1) mutually orthogonal g i − halfstarter graphs Ten, we obtain k mutually orthogonal h i − half-starter graphs G h i � G f i ⊗ G g i , which are represented by the functions h i (xy) � ((i + 1)x)g i (y).Since h  p(xy) − h p (xy) � ((p + 1)x)g p (y) − ((q + 1)x)g q (y) � ((p + 1)x − (q + 1)x)(g p (y) − g q (y)): p, q ∈ Z k , p ≠ q} � Z m ⊗ Z n , then G hp and G hq are orthogonal.Te edge set of the graphs G h i can be obtained as follows, since (g i  (y)))}.Now, we want to prove the isomorphism of the two graphs G hp and G hq .For p, q ∈ Z k , p ≠ q, we have , 2), then for G hp � G f p ⊗ G g p and G hq � G f q ⊗ G g q , the degree vector of G hp is equal to the degree vector of G hq .Tis means that , 2).Hence, G hp and G hq are isomorphic.
, which are represented by the functions g i (y) � y 2 + iy, y ∈ Z 9 , i ∈ Z 3 (ingredient (iv)).Ten, we obtain 3 mutually orthogonal , then G hp and G hq are orthogonal.Te edge set of the graphs G h i can be obtained as follows, since Now, we want to prove the isomorphism of the two graphs G hp Table 7: Te edge set of the graphs G g 0 , G g 1 , and G g 2 for Example 5.

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Example 5. Let f i (x) � x 2 + ix be f i − half-starter graphs G f i of K 3,3 for all i, x ∈ Z 3 and g i (y) � y 2 + iy be g i − half-starter graphs G g i of K 5,5 for all i, x ∈ Z 5 .Since min 3, 5 { } � 3, then we obtain 3 h i − half-starter graphs G h i of K 15,15 , which are represented by the functions h i (xy) � f i (x)g i (y) for all i ∈ Z 3 .Te edge sets of G f i , G g i , and G h i are shown in Tables 6, 7, and 8, where . Also, the three MOGS corresponding to the functions f i (x) are A 0 , A 1 , and A 2 , the three MOGS corresponding to the functions g i (y) are B 0 , B 1 , and B 2 , and the three MOGS corresponding to the functions h i (xy) are C 0 , C 1 , and C 2 .See Figures 4, 5, and 6.
In the following section, we present the general tensor product technique for constructing the MOGS.As stated above, the MOGS represent mutually orthogonal covers (MOCs) of complete bipartite graphs.A k mutually orthogonal covers (k MOCs) of the complete bipartite graph K n,n by F is a family G of isomorphic copies of a given subgraph F such that they cover every edge of K n,n k times and the intersection of any two of them contains at most one edge.

Tensor Products of MOCs
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Discussion
All the results in this paper are based on recursive construction techniques as stated above.In the literature, the Kronecker product of graph squares has been used to construct some results for MOGS.Herein, we defned three novel product techniques, which are the Cartesian product of half-starters' vectors, the half-starters' function product, and the graph tensor product.Some graphs can be represented by vectors, so the Cartesian product can be used with this class of graphs.Other graphs cannot be represented by vectors but can be represented by functions; hence, the function product can be used with this class of graphs.In addition, there is a third class of graphs that cannot be represented by vectors and functions; in this case, the tensor product of graphs is applied to construct the MOGS.All the results from the literature of MOGS with small orders can be used to get MOGS with higher orders by applying the new product techniques defned in this paper.Te main results are Propositions 1, 2, and 3, which introduce the construction techniques based on the defned novel product techniques.All the remaining results in the paper are direct applications to these propositions.Tese results are MOGS for disjoint unions of stars such as nK 1,1 ∪ (((((m − 1))n))/2) K 1,2 ,(n − 2)K 1,1 ∪ ((((m − 1))(n − 2) + 2)/2)K 1,2 ∪ ((m − 1) /2)K 1,4 ,K 1,3 ∪ 3 K 1,2 ∪ (((m − 1))/2)K 1,6 ∪ (3(((m − 1)) /2)) K 1,4 ,m(n − 2)K 1,1 ∪ mK 1,2 , and 3K 1,1 ∪ 3n + 1/2K 1,2 ∪ (n − 1) K 1,4 .All the constructed results in this paper can be used to generate new graph-orthogonal arrays, new graph-authentication codes, and new graph-transversal designs [3,23].Tey can also be used in the design of experiments [24].

Conclusion
In conclusion, we can say that the proposed novel product techniques are helping tools for constructing several new results concerned with the MOGS that have not been constructed before.It is clear that the proposed product techniques cannot be used to construct MOGS with prime order.In future work, we will try to fnd new recursive construction techniques for the MOGS.

Figure 2 :
Figure 2: Degrees of vertices for the graph G.
Let A and B be simple graphs, then the tensor product, A × B, of A and B, is the graph with the vertex set V(A) × V(B) and the edge set E(A × B) � (a, b)(c, d): ac ∈ E(A) { } and b d ∈ E(B).If the simple graphs A and B are bipartite with bipartitions (E, F) and (Y, Z), respectively, then the induced subgraphs (

Table 1 :
Te nomenclature used in the paper.

Table 2 :
Te used vectors in Example 3.

Table 4 :
Te graph G f 1 and it's translates for Example 4.

Table 5 :
Te graph G f 2 and it's translates for Example 4.

Table 3 :
Te graph G f 0 and it's translates for Example 4.
2 are represented by the functions f i (x) � x 2 + ix, where i, x ∈ Z n .(ii) Let n be a prime number.Ten, N(n, nK 1,1 ) � n − 1 and the f i − half-starter graphs G f i � nK 1,1 are represented by the functions

Table 6 :
Te edge set of the graphs G f 0 , G f 1 , and G f 2 for Example 5.

Table 8 :
Te edge set of the graphs G h 0 , G h 1 , and G h 2 for Example 5.