Orthogonal double covers of complete graphs by fat catepillars

An orthogonal double cover (ODC) of the complete graph Kn by some graph G is a collection of n spanning subgraphs of Kn, all isomorphic to G, such that any two of the subgraphs share exactly one edge and every edge of Kn is contained in exactly two of the subgraphs. A necessary condition for such an ODC to exist is that G has exactly n− 1 edges. We show that for any given positive integer d, almost all caterpillars of diameter d admit an ODC of the corresponding complete graph.