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Nowadays, graph theory is one of the most exciting fields of mathematics due to the tremendous developments in modern technology, where it is used in many important applications. The orthogonal double cover (O D C) is a branch of graph theory and is considered as a special class of graph decomposition. In this paper, we decompose the complete bipartite graphs K x , x by caterpillar graphs using the method of O D C s. The article also deals with constructing the O D C s of K x , x by general symmetric starter vectors of caterpillar graphs such as stars–caterpillar, the disjoint copies of cycles–caterpillars, complete bipartite caterpillar graphs, and the disjoint copies of caterpillar paths. We decompose the complete bipartite graph by the complete bipartite subgraphs and by the disjoint copies of complete bipartite subgraphs using general symmetric starter vectors. The advantage of some of these new results is that they enable us to decompose the giant networks into large groups of small networks with the comprehensive coverage of all parts of the giant network by using the disjoint copies of symmetric starter subgraphs. The use case of applying the described theory for various applications is considered. [ABSTRACT FROM AUTHOR]

An orthogonal double cover (ODC) of a graph H is a collection G={Gv:v∈V(H)} of |V(H)| subgraphs of H such that every edge of H is contained in exactly two members of G and for any two members Gu and Gv in G, |E(Gu)∩E(Gv)| is 1 if u and v are adjacent in H and it is 0 if u and v are nonadjacent in H. In this paper, we are concerned with the Cartesian product of symmetric starter vectors of orthogonal double covers of the complete bipartite graphs and using this method to construct ODCs for new graph classes.

Let H be a graph on n vertices and G a collection of n subgraphs of H, one for each vertex, G is an orthogonal double cover (ODC) of H if every edge of H occurs in exactly two members of G and any two members share an edge whenever the corresponding vertices are adjacent in H and share no edges whenever the corresponding vertices are nonadjacent in H. In this paper, we are concerned with symmetric starter vectors of the orthogonal double covers (ODCs) of the complete bipartite graph and using the method of cartesian product of symmetric starter vectors to construct ODC of the complete bipartite graph by G, where G is a complete bipartite graph, disjoint union of different complete bipartite graphs and disjoint union of finite copies of a complete bipartite graph.

Let H be a graph on n vertices and G a collection of n subgraphs of H , one for each vertex, G is an orthogonal double cover (ODC) of H if every edge of H occurs in exactly two members of G and any two members share an edge whenever the corresponding vertices are adjacent in H and share no edges whenever the corresponding vertices are nonadjacent in H . In this paper, we are concerned with symmetric starter vectors of the orthogonal double covers (ODCs) of the complete bipartite graph and using the method of cartesian product of symmetric starter vectors to construct ODC of the complete bipartite graph by G , where G is a complete bipartite graph, disjoint union of different complete bipartite graphs and disjoint union of finite copies of a complete bipartite graph. [ABSTRACT FROM AUTHOR]

Abstract: Let and be graphs, such that is isomorphic to a subgraph of . An orthogonal double cover (ODC) of by is a collection of subgraphs of , all isomorphic with , such that (i) every edge of occurs in exactly two members of and (ii) and share an edge if and only if and are adjacent in . The main question is: given the pair , is there an ODC of by ? An obvious necessary condition is that is regular. A technique to construct ODCs for Cayley graphs is introduced. It is shown that for all where is a 3-regular Cayley graph on an abelian group there is an ODC, a few well known exceptions apart. [Copyright &y& Elsevier]

In this paper, we generalize the orthogonal double covers (ODC) of K n , n as follows. The circular intensely orthogonal double cover design (CIODCD) of X = K n , n , ... , n ︸ m is defined as a collection T = { G 0 0 , G 1 0 , ... , G (n − 1) 0 } ∪ { G 0 1 , G 1 1 , ... , G (n − 1) 1 } of isomorphic spanning subgraphs of X such that every edge of X appears twice in the collection T , E (G i 0 ) ∩ E (G j 0 ) = E (G i 1 ) ∩ E (G j 1 ) = 0 , i ≠ j and   E (G i 0 ) ∩ E (G j 1 ) = λ = m 2 , i , j ∈ ℤ n. We define the half starters and the symmetric starters matrices as constructing methods for the CIODCD of X. Then, we introduce some results as a direct application to the construction of CIODCD of X by the symmetric starters matrices. [ABSTRACT FROM AUTHOR]