Construction of the enormous complete bipartite graphs and orthogonal double covers based on the Cartesian product

Let K be a graph with n vertices and G= { π(x) : x ∈v (K)} be a collection of isomorphic pages (called subgraphs) of K . Then G is an orthogonal double cover (ODC) of K by G iff (i) every edge of K is repeated exactly twice in G and (ii) π(a) and π(b) have one edge iff a and b are adjacent vertices in K . In this paper, we are interested in finding new symmetric starter vectors (SSV) based on Cartesian product methods, such as SSV of complete bipartite graphs with complete bipartite graphs, disjoint copies of paths with stars, stars with caterpillars, disjoint copies of stars with caterpillars, disjoint copies of cycles with caterpillars, disjoint copies of paths with caterpillars, and complete bipartite graphs with caterpillars. Then we use these symmetric starter vectors to get the enormous graphs and construct the ODCs.