Decompositions of Complete Multigraphs Related to Hadamard Matrices

Let bp(μKv) be the minimum number of complete bipartite subgraphs needed to partition the edge set ofμKv, the complete multigraph withμedges between each pair of itsvvertices. Many papers have examined bp(μKv) forv⩾2μ. For eachμandvwithv⩽2μ, it is shown here that if certain Hadamard and conference matrices exist, then bp(μKv) must be one of two numbers. Also, generalizations to decompositions and covers by completes-partite subgraphs are discussed and connections to designs and codes are presented.