Decompositions of Complete Multigraphs Related to Hadamard Matrices

Abstract

Let bp(+K v ) be the minimum number of complete bipartite subgraphs needed to partition the edge set of +K v , the complete multigraph with + edges between each pair of its v vertices. Many papers have examined bp(+K v ) for v 2+. For each + and v with v 2+, it is shown here that if certain Hadamard and conference matrices exist, then bp(+K v ) must be one of two numbers. Also, generalizations to decompositions and covers by complete s-partite subgraphs are discussed and connections to designs and codes are presented. Academic PressThroughout the paper, +K v denotes the complete multigraph with v 2 vertices and + edges between each pair of distinct vertices. A biclique in +K v is a simple complete bipartite subgraph. A biclique decomposition of +K v is a collection of bicliques whose edge sets partition the edge set of +K v . The biclique decomposition number of +K v , denoted bp(+K v ), is the minimum number of bicliques needed in a biclique decomposition of +K v . Minimum biclique decompositions of #K v and of +K v together give a biclique decomposition of (#++) K v . Therefore, bp((#++) K v ) bp(#K v )+bp(+K v ).(1)