Orthogonal Double Covers of Complete Bipartite Graphs by A Special Class of Disjoint Union of Path and A Complete Bipartite Graph

Let H be a graph on n vertices and let G be 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 is contained 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 non-adjacent in H. In this paper, we are concerned with the symmetric starter vectors of the orthogonal double covers of the complete bipartite graphs and using this method to construct ODCs by the disjoint union of path and a complete bipartite graph. Here, we consider Pm the path on m vertices where 4 ≤ m ≤ 11: