Abstract
For every fixedk≥3 there exists a constantc k with the following property. LetH be ak-uniform,D-regular hypergraph onN vertices, in which no two edges contain more than one common vertex. Ifk>3 thenH contains a matching covering all vertices but at mostc k ND −1/(k−1). Ifk=3, thenH contains a matching covering all vertices but at mostc 3 ND −1/2ln3/2 D. This improves previous estimates and implies, for example, that any Steiner Triple System onN vertices contains a matching covering all vertices but at mostO(N 1/2ln3/2 N), improving results by various authors.
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Research supported in part by a USA-Israel BSF grant.
Research supported in part by a USA-Israel BSF Grant.
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Alon, N., Kim, JH. & Spencer, J. Nearly perfect matchings in regular simple hypergraphs. Isr. J. Math. 100, 171–187 (1997). https://doi.org/10.1007/BF02773639
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DOI: https://doi.org/10.1007/BF02773639