Summary
A bipartite covering of order k of the complete graph K n on n vertices is a collection of complete bipartite graphs so that every edge of K n 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 \(\Theta (k{n}^{1/k})\). 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.
Research supported in part by the Sloan Foundation, Grant No. 93-6-6.
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Alon, N. (2013). Neighborly Families of Boxes and Bipartite Coverings. In: Graham, R., Nešetřil, J., Butler, S. (eds) The Mathematics of Paul Erdős II. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-7254-4_2
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