Decompositions of complete multigraphs derived from Hadamard matrices

Let b(μKv) be the minimum number of complete bipartite subgraphs needed to partition the edge set of μKv, the complete multigraph with μ edges between each pair of vertices. Previous work by Gregory and Vander Meulen determined that for μ odd with v⩽2μ, and subject to the existence of certain Hadamard and conference matrices, then b(μKv) is one of two numbers. By considering forbidden submatrices of a vertex–biclique incidence matrix, we determine conditions for when the lower of these numbers is not attained, and describe constructions that show the lower bound can be attained in the remaining cases. Assuming the standard necessary conditions for the existence of Hadamard and conference matrices are sufficient, this completes the determination of b(μKv) for all μ and v such that v⩽2μ.