Neighborly Families of Boxes and Bipartite Coverings
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 Θ(kn 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.
Unable to display preview. Download preview PDF.
- 2.N. Alon, R. A. Brualdi and B. L. Shader, Multicolored forests in bipartite decompositions of graphs, J. Combinatorial Theory, Ser. B (1991), 143–148.Google Scholar
- 3.N. G. de Bruijn and P. Erdős, On a combinatorial problem, Indagationes Math. 20 (1948), 421–423.Google Scholar
- 4.P. Erdős, On sequences of integers none of which divides the product of two others, and related problems, Mitteilungen des Forschungsinstituts für Mat. und Mech., Tomsk, 2 (1938), 74–82.Google Scholar
- 5.P. Erdős and G. Purdy, Some extremal problems in combinatorial geometry, in: Handbook of Combinatorics (R. L. Graham, M. Grötschel and L. Lovász eds.), North Holland, to appear.Google Scholar
- 6.A. Felzenbaum and M. A. Perles, Private communication.Google Scholar
- 9.R. L. Graham and H. O. Pollak, On embedding graphs in squashed cubes, In: Lecture Notes in Mathematics 303, pp 99–110, Springer Verlag, New York-Berlin Heidelberg, 1973.Google Scholar
- 10.J. Kasem, Neighborly families of boxes, Ph. D. Thesis, Hebrew University, Jerusalem, 1985.Google Scholar
- 12.J. Pach and P. Agarwal, Combinatorial Geometry, DIM ACS Tech. Report 41–51, 1991 (to be published by J. Wiley).Google Scholar