Abstract
Sets of mutually orthogonal Latin squares prescribe the order in which to apply different treatments in designing an experiment to permit effective statistical analysis of results, they encode the incidence structure of finite geometries, they encapsulate the structure of finite groups and more general algebraic objects known as quasigroups, and they produce optimal density error-correcting codes. This 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 find the minimum number of mutually orthogonal subgraphs for certain graphs based on the proposed product techniques. As a direct application to the proposed different 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.
1. 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 different species in an environment, to represent who influences whom in an organization, and to represent the outcomes of round-robin 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 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.
Definition 1. (see [4]). Let be a subgraph of with size A square matrix of order is an square if every element in is found exactly times in , and the graphs with are isomorphic to The elements of are used for labeling the rows of , and the elements of are used for labeling the columns of An edge decomposition of by a graph can be represented by an square.
Definition 2. (see [4]). Suppose is an square of order with entries from a set , and is an square of order with entries from a set Then, the two squares and are orthogonal if, for every and for every there exists exactly one cell such that and A set of squares of order say are called pairwise orthogonal (mutually orthogonal) squares (MOGS) if and are orthogonal for all Here, we consider
Theorem 1. For the bipartite graph having edges, denotes the maximum number in a largest possible set of MOGS of by For every bipartite graph with edges, we have
Great efforts have been made to get the solution to several problems concerned with the MOLS since Euler first asked about MOLS to solve the thirty-six officer’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 efforts have been concentrated on refining and finding novel applications for these approaches. The authors of [8] proposed an integrated firefly algorithm based on MOLS, named FA-MOLS, to address the quadratic assignment problem. Liu [9] introduced the packing of Latin squares by BCL algebras. The 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]. The 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, where is a path with vertices and is a prime number. Sampathkumar et al. [17] solved this conjecture. El-Shanawany [18] found . El-Shanawany [19] computed where F is disjoint copies of some subgraphs of 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 graph-authentication codes in [3, 23]. MOGS are used to construct orthogonal arrays that have many applications [24].
The 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 different product techniques to MOGS that yield new MOGS results. The proposed product techniques are the half-starters’ vectors Cartesian product, half-starters’ function product, and graph tensor product. The novelty of the current paper is demonstrated by the fact that it is the first 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 find the minimum number of mutually orthogonal subgraphs for certain graphs based on the proposed product techniques. As a direct application to the proposed different product techniques, mutually orthogonal graph squares for disjoint unions of stars are constructed. The main difference 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 half-starters’ function product is applied to the functions, and the graph tensor product is applied to graphs.
The remaining part of the present paper is divided as follows: Section 2 is devoted to MOGS from mutually orthogonal half-starters’ 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.
2. MOGS from Mutually Orthogonal Half Starters’ Vectors
If we have a graph which is considered a subgraph of with edges, then the graph is called the translate of and If the edge then its length is defined by where arithmetic operations are calculated modulo A graph is called a half-starter w.r.t. if and the edges in have different lengths that are equivalent to the group
Theorem 2. (see [9]). If is a half-starter, then an edge decomposition of can be constructed by finding all the translates of and taking their union; that is,
The vector can be used to represent the half-starter where and is the unique vertex that belongs to the unique edge of length in with Two half-starters’ vectors and are said to be orthogonal if A set of half-starters’ vectors is mutually orthogonal if and are orthogonal for every It is worth noting that each half-starter and its translates of a subgraph of are equivalent to square. Hence, the set of mutually orthogonal half-starters and their translates are equivalent to a set of mutually orthogonal squares.
3. MOGS Based on the Cartesian Product of Half-Starters’ Vectors
The Cartesian product of half-starters’ vectors has been defined in literature for constructing orthogonal double covers of This method has been applied to construct orthogonal double covers of by new graph classes. The Cartesian product of two vectors corresponding to two half-starter graphs is considered a very special case of the tensor product of these two half-starter graphs.
Definition 3. The tensor product of two graphs and is defined as follows. If a vertex is adjacent to a vertex in and a vertex is adjacent to a vertex in then the vertex is adjacent to the vertex in
Example 1. Figure 1 exhibits an example of the graphs and
Definition 4. (see [25]). Let be a graph, and belongs to the vertex set of The number of edges incident at in is called the degree (or valency) of the vertex in and is denoted by From the degrees of vertices of we can construct a sequence which is called a degree sequence of when the vertices are taken in the same order. It is customary to put this sequence in nondecreasing or nonincreasing order. This gives a unique sequence.
Example 2. (see [25]). In the graph of Figure 2, the number within the parentheses indicates the degree of the corresponding vertex. The degree sequence of is
Definition 5. If we have the vector then, by determining the repetition number of each element in the vector we get the vector where is the repetition number of the element By the ascending order for the vector we get the degree sequence of the vector defined by where
Definition 6. If we have the two vectors and the two vectors where then the two half-starters and are isomorphic if and or and where is the degree sequence of the vector is the degree sequence of the vector is the degree sequence of the vector and is the degree sequence of the vector In our paper, we consider the case of and for the isomorphism of the two half-starters and
For Proposition 1, if we have the two half-starters and which are represented by the vectors and respectively, then the graph is defined by the edge set
Proposition 1. If there are mutually orthogonal half-starters’ vectors of length for the graph and mutually orthogonal half-starter’ vectors of length for the graph then there are mutually orthogonal half-starters’ vectors of length for the graph
Proof 1. For let be mutually orthogonal half-starters’ vectors of length and be mutually orthogonal half-starters’ vectors of length where and . Hence, Then, where For we conclude from (1) and (2),Hence, from (3), the two half-starters and are orthogonal. Since and thenNow, we will try to prove the isomorphism of the graphs , We have and Since the degree sequence of the vector equals the degree sequence of the vector the degree sequence of the vector equals the degree sequence of the vector the degree sequence of the vector equals the degree sequence of the vector and the degree sequence of the vector equals the degree sequence of the vector then the degree sequence of the vector equals the degree sequence of the vector and the degree sequence of the vector equals the degree sequence of the vector Hence, the two half-starters and are isomorphic.
Example 3. Let be a prime Then, we have mutually orthogonal half-starters’ vectors defined by where and see [11]. Hence, we have 3 mutually orthogonal half-starters’ vectors for both and which are and respectively. Then, we obtain 3 mutually orthogonal half-starters’ vectors as shown in Table 2. See Figure 3.
Also, the three MOGS corresponding to the vectors are and the three MOGS corresponding to the vectors are and and the three MOGS corresponding to the vectors are and
Of course in and one can easily replace the ordered pairs 00, 01, 02, 10, 11, 12, 20, 21, 22, 30, 31, 32, 40, 41, 42 by 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 to obtain three mutually orthogonal -squares of order 15, where their elements are the usual symbols.
4. MOGS from Mutually Orthogonal Half-Starters’ Functions
In what follows, if and then the graphs will be represented by the functions The edge set of the graphs is Every graph from the graphs represents unions of stars which have the same direction where every vertex belongs to has a degree of one, that is, The graph is called half-starter if where is a subgraph of with edges and
Definition 7. (see [9]). Let Then, the graph is called the translate of
Remark 1. (see [9]). The union of all translates of forms an edge decomposition of that is,
Definition 7 and Remark 1 show that every half-starter graph and the translates are equivalent to -square.
Let be subgraphs of with Then, the set is called a set of mutually orthogonal subgraphs, if for all and If the two half-starters and are orthogonal, then the two sets of translates of and are orthogonal. A set of edge decompositions is a set of MOGS if and are orthogonal for all and
Example 4. Let be half-starter graphs of for all The edge set of the graphs and their translates are shown in Tables 3, 4, and 5.
Hence, we deduce the following three mutually orthogonal -squares:
Definition 8. For the half-starter graph represented by the function where and we have for all By determining the degree of each vertex belonging to from we get the vector where is the degree of the vertex By the ascending order for the vector we get the degree vector of denoted by where
Definition 9. If we have the f-half-starter graph represented by the function and the -half-starter graph represented by the function then and are isomorphic if where is the degree vector of and is the degree vector of
We shall denote by the maximal number of half-starter graphs in the largest possible set of mutually orthogonal subgraphs in Hereafter, if we have where and having edges and where and having edges and then we obtain where and by Proposition 2. Also, we present some results as direct applications to Proposition 2. In the following, if there is no danger of ambiguity, if we can write as
Proposition 2. If where and have edges, where and have edges, and then where are isomorphic graphs having edges.
Proof 2. Let half-starter graphs be represented by the functions and half-starter graphs be represented by the functions Then, we obtain the half-starter graphs which are represented by the functionsSince then and are orthogonal. The edge set of the graphs can be obtained as follows, since and then Now, we want to prove the isomorphism of the two graphs and For we have and then for and the degree vector of is equal to the degree vector of This means that Hence, and are isomorphic.
All the following results are based on (i) Proposition 2 and (ii) the following ingredients (see [9]).(i)Let be a prime number. Then, and the half-starter graphs are represented by the functions where (ii)Let be a prime number. Then, and the half-starter graphs are represented by the functions where (iii)Let be a prime number. Then, and the half-starter graphs are represented by the functions (iv)Let Then, and the half-starter graphs are represented by the functions where (v)Let Then for the half-starter graphs are represented by the functionsThese known half-starter graphs are the ingredients for the following results. These ingredients are some of the literature results and are not all of the literature results.
Theorem 3. Let be odd primes. Then, where and
Proof 3. We have mutually orthogonal half-starter graphs which are represented by the functions where (ingredient (i)) and mutually orthogonal half-starter graphs which are represented by the functions where (ingredient (ii)). Then, we obtain mutually orthogonal half-starter graphs which are represented by the functions Since then and are orthogonal. The edge set of the graphs can be obtained as follows, since and then Now, we want to prove the isomorphism of the two graphs and For we have and then for and the degree vector of is equal to the degree vector of This means that . Hence, and are isomorphic.
Theorem 4. Let be primes. Then, where and
Proof 4. We have mutually orthogonal half-starter graphs which are represented by the functions where (ingredient (i)) and mutually orthogonal half-starter graphs which are represented by the functions (ingredient (iii)). Then, we obtain mutually orthogonal half-starter graphs which are represented by the functions Since then and are orthogonal. The edge set of the graphs can be obtained as follows, since and then Now, we want to prove the isomorphism of the two graphs and For we have and then for and the degree vector of is equal to the degree vector of This means that Hence, and are isomorphic.
Theorem 5. Let be odd prime. Then, where
Proof 5. We have mutually orthogonal half-starter graphs which are represented by the functions where (ingredient (i)) and mutually orthogonal half-starter graphs which are represented by the functions (ingredient (iv)). Since then we obtain mutually orthogonal half-starter graphs which are represented by the functions Since then and are orthogonal. The edge set of the graphs can be obtained as follows, since and then Now, we want to prove the isomorphism of the two graphs and For we have and then for and the degree vector of is equal to the degree vector of This means that Hence, and are isomorphic.
Theorem 6. Let be prime. Then, where
Proof 6. We have mutually orthogonal half-starter graphs which are represented by the functions where (ingredient (i)) and mutually orthogonal half-starter graphs which are represented by the functionswhere (ingredient (v)). Since then we obtain mutually orthogonal half-starter graphs which are represented by the functions Since then and are orthogonal. The edge set of the graphs can be obtained as follows, since and then Now, we want to prove the isomorphism of the two graphs and For we have and then for and the degree vector of is equal to the degree vector of This means that Hence, and are isomorphic.
Theorem 7. Let be primes. Then where and
Proof 7. We have mutually orthogonal half-starter graphs which are represented by the functions where (ingredient (ii)) and mutually orthogonal half-starter graphs which are represented by the functions (ingredient (iii)). Then, we obtain mutually orthogonal half-starter graphs which are represented by the functions Since then and are orthogonal. The edge set of the graphs can be obtained as follows, since and then Now, we want to prove the isomorphism of the two graphs and For we have and then for and the degree vector of is equal to the degree vector of This means that Hence, and are isomorphic.
Theorem 8. Let be prime. Then where
Proof 8. We have mutually orthogonal half-starter graphs which are represented by the functions (ingredient (iii)) and mutually orthogonal half-starter graphs which are represented by the functions (ingredient (iv)). Then, we obtain mutually orthogonal half-starter graphs which are represented by the functions Since then and are orthogonal. The edge set of the graphs can be obtained as follows, since and then Now, we want to prove the isomorphism of the two graphs and For we have and , then for and the degree vector of is equal to the degree vector of This means that Hence, and are isomorphic.
Example 5. Let be half-starter graphs of for all and be half-starter graphs of for all Since then we obtain 3 half-starter graphs of which are represented by the functions for all The edge sets of and are shown in Tables 6, 7, and 8, where and Also, the three MOGS corresponding to the functions are and the three MOGS corresponding to the functions are and and the three MOGS corresponding to the functions are and 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 mutually orthogonal covers ( MOCs) of the complete bipartite graph by is a family of isomorphic copies of a given subgraph such that they cover every edge of times and the intersection of any two of them contains at most one edge.
5. Tensor Products of MOCs
Let and be simple graphs, then the tensor product, , of and , is the graph with the vertex set and the edge set and . If the simple graphs and are bipartite with bipartitions and , respectively, then the induced subgraphs and are called the weak-tensor products of and . We denote the weak-tensor product by .
Proposition 3. If there are MOCs of by and MOCs of by , then there are MOCs of by
Proof 9. Let where be MOCs of by on where and is the bipartition of and let where be MOCs of by on where and is the bipartition of Let and the partite sets of be and . Consider the set of subgraphs of . Clearly,
Claim 1. Every edge of occurs in exactly graphs of .
Consider an arbitrary edge of Since and are MOCs of by and MOCs of by respectively, the edges and are, respectively, in exactly graphs of and . Let the graphs containing be and that of be . Then, the graphs containing the edge are
Claim 2. Let . Any two graphs in have no edges in common.
The two graphs and have no edges in common, because and .
Claim 3. Any graph in and any graph in have exactly one edge in common, .
The two graphs and have exactly one edge in common, since and .
By Claims and is MOCs of by
6. MOCs of Complete Bipartite Graphs Based on Tensor Product
All the following results are based on the tensor product in Proposition 3 and the existence of MOCs for some classes of graphs. These graphs can be used as ingredients for the tensor product to obtain new MOCs. See [9] for the ingredients from to Addition and subtraction are calculated modulo for the following ingredients.(i)Let where be MOCs of by be a prime (ii)Let where be MOCs of by be a prime (iii)If , then the 3 MOCs of by where .(iv)If , then MOCs of by where for (v)If , then MOCs of by (follows from Theorem 1 in [4], by setting and where
These known MOCs are the ingredients for the tensor product to obtain the following results. Note that we used some of the ingredients from the literature.
Theorem 9. Let be odd primes and . Then, there are MOCs of by .
Proof 10. We have MOCs of by (ingredient ()) and MOCs of by (ingredient ()) If then we construct MOCs of by (Proposition 3)
Theorem 10. Let be odd prime. Then, there are MOCs of by .
Proof 11. We have MOCs of by (ingredient ()) and MOCs of by (ingredient ()) Then, we construct MOCs of by (Proposition 3)
Theorem 11. Let be odd prime. Then, there are MOCs of by .
Proof 12. We have MOCs of by (ingredient ()) and MOCs of by (ingredient ()) Then, we construct MOCs of by (Proposition 3).
Theorem 12. Let be odd prime. Then, there are MOCs of by .
Proof 13. We have MOCs of by (ingredient ()) and MOCs of by (ingredient ()) Then, we construct MOCs of by (Proposition 3).
Theorem 13. Let be odd prime. Then, there are MOCs of by .
Proof 14. We have MOCs of (ingredient ()) and MOCs of ( (ingredient ()). Then, we construct MOCs of by (Proposition 3)
Example 6. We have mutually orthogonal covers of by shown in Figure 7, and mutually orthogonal covers of by shown in Figure 8. Hence, we construct mutually orthogonal covers of by shown in Figure 9.
7. 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 defined 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 defined in this paper. The main results are Propositions 1, 2, and 3, which introduce the construction techniques based on the defined novel product techniques. All the remaining results in the paper are direct applications to these propositions. These results are MOGS for disjoint unions of stars such as and . 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]. They can also be used in the design of experiments [24].
8. 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 find new recursive construction techniques for the MOGS.
Data Availability
The data used to support the findings of this study are available from the corresponding author on request.
Conflicts of Interest
The authors declare no conflict of interest.