Smaller embeddings of partial k-star decompositions

A k-star is a complete bipartite graph K1,k. For a graph G, a k-star decomposi-tion of G is a set of k-stars in G whose edge sets partition the edge set of G. If we weaken this condition to only demand that each edge of G is in at most one k-star, then the resulting object is a partial k-star decomposition of G. An embedding of a partial k-star decomposition A of a graph G is a partial k-star decomposition B of another graph H such that A subset of B and G is a subgraph of H. This paper considers the problem of when a partial k-star decomposition of Kn can be embedded in a k -star decomposition of Kn+3 for a given integer s. We improve a result of Noble and Richardson, itself an improvement of a result of Hoffman and Roberts, by showing that any partial k-star decomposition of Kn can be embedded in a k-star decom-position of Kn+3 for some s such that s < 94k when k is odd and s < (6 - 2 root 2)k when k is even. For general k, these constants cannot be improved. We also obtain stronger results subject to placing a lower bound on n.