Orthogonal double covers of complete graphs by lobsters of diameter 5

An orthogonal double cover (ODC) of the complete graph K_n by a graph G is a collection G = {G_i i = 1, 2,..., n} of spanning subgraphs of K_n, all isomorphic to G, with the property that every edge of K_n belongs to exactly two members of G and any two distinct members of G share exactly one edge. A lobster of diameter five is a tree arising from a double star by attaching any number of pendant vertices to each of its vertices of degree one. We show that for any double star R(p, q) there exists an ODC of K_n by all lobsters of diameter five (with finitely many possible exceptions) arising from R(p, q).