Balls, bins, and embeddings of partial k-star designs

Define a k-star to be the complete bipartite graph K-1,K-k. In a 2014 article, Hoffman and Roberts prove that a partial k-star decomposition of K-n can be embedded in a k-star decomposition of k(n+s) where s is at most 7k - 4 if k is odd and 8k - 4 if k is even. In our work, we offer a straightforward construction for embedding partial k-star designs and lower these bounds to 3k - 2 and 4k - 2, respectively. (C) 2019 Elsevier B.V. All rights reserved.