Embedding partial 3-star designs

Define a 3-star decomposition of K n as being a collection of subgraphs, each isomorphic to K 1 , 3 , with the property that each edge of K n appears in exactly one of the subgraphs. A partial 3-star decomposition is similarly defined except each edge appears in at most one of the subgraphs. In this work, it is shown that any partial 3-star decomposition of K n can be embedded into a decomposition of K n + s where s ≤ 4 . Furthermore, we determine, for any maximal partial 3-star decomposition P of K n , the minimum s ∈{ 1 , 2 , 3 , 4 } such that P can be embedded into a decomposition of K n + s .