On covering graphs by complete bipartite subgraphs

We prove that, if a graph with ee edges contains mm vertex-disjoint edges, then m2/em2/e complete bipartite subgraphs are necessary to cover all its edges. Similar lower bounds are also proved for fractional covers. For sparse graphs, this improves the well-known fooling set lower bound in communication complexity. We also formulate several open problems about covering problems for graphs whose solution would have important consequences in the complexity theory of boolean functions.