Balanced star decompositions of regular multigraphs and λ-fold complete bipartite graphs

Let Sk denote the star with k edges. A balanced Sk-decomposition of a multigraph G is a family D of subgraphs of G whose edge sets form a partition of the edge set of G such that each member of D is isomorphic to Sk, and every vertex of G belongs to the same number of members in D. In this paper, we obtain the following results: