Neighborly families of boxes and bipartite coverings

Abstract

A bipartite covering of order k of the complete graph Kn on n vertices is a collection of complete bipartite graphs so that every edge of Kn lies in at least 1 and at most k of them. It is shown that the minimum possible number of subgraphs in such a collection is Θ(kn1/k). This extends a result of Graham and Pollak, answers a question of Felzenbaum and Perles, and has some geometric consequences. The proofs combine combinatorial techniques with some simple linear algebraic tools.