Abstract | Let H be a graph on n vertices and C a collection of n subgraphs of H, one for each vertex. Then C is an orthogonal double cover (ODC) of H if every edge of H occurs in exactly two members of C and any two members of C share exactly an edge whenever the corresponding vertices are adjacent in H. If all subgraphs in C are isomorphic to a given graph G, then C is said to be an ODC of H by G.
We construct the ODCs of the complete bipartite graph with two classes of n vertices by a
graph G which is the union of a m-path and
a (n-m)-star, where the center of the star is a one of the path ends, for all m=5,6,7,8,9,10. In all cases, G is a symmetric starter of the cyclic group of order n. |