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 \( \mathcal{G} = \{G_i \mid i=1,2,\dots,n\} \) 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 \( \mathcal{G} \) and any two distinct members of \( \mathcal{G} \) 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 \( R(p, q) \).