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

AbstractLet 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:(1)A necessary and sufficient condition for an r-regular multigraph to have a balanced Sk-decomposition.(2)A necessary and sufficient condition for the λ-fold complete bipartite graph λKm,n to have a balanced Sk-decomposition