Geodesic
Orthogonal double covers of complete graphs by fat caterpillars
Discussiones Mathematicae. Graph Theory, Tome 26 (2006) no. 2, pp. 343-349

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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.
Keywords: ODC, orthogonal double cover, graph decomposition, self-orthogonal factorization 
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Froncek, Dalibor; Leck, Uwe. Orthogonal double covers of complete graphs by fat caterpillars. Discussiones Mathematicae. Graph Theory, Tome 26 (2006) no. 2, pp. 343-349. http://geodesic.mathdoc.fr/item/DMGT_2006_26_2_a14/