## A note on perfect matchings in uniform hypergraphs with large minimum collective degree

### Authors:

- Vojtêch Rödl,
- Andrzej Ruciński,
- Mathias Schacht,
- Endre Szemerédi

### Abstract

For an integer k ≥ 2 and a k-uniform hypergraph H, let δ_{k-1}(H) be the largest integer d such that every (k -1)-element set of vertices of H belongs to at least d edges of H. Further, let t(k, n) be the smallest integer t such that every k-uniform hypergraph on n vertices and with δ_{k-1}(H) ≥ t contains a perfect matching. The parameter t(k, n) has been completely determined for all k and large n divisible by k by Rödl, Ruciński, and Szemerédi in [Perfect matchings in large uniform hypergraphs with large minimum collective degree, submitted]. The values of t(k, n) are very close to n/2-k. In fact, the function t(k, n) = n/2 - k + c_{n,k}, where c_{n,k} ∈ (3/2, 2, 5/2, 3) depends on the parity of k and n. The aim of this short note is to present a simple proof of an only slightly weaker bound: t(k, n) ≤ n/2 + k/4. Our argument is based on an idea used in a recent paper of Aharoni, Georgakopoulos, and Sprüssel.

- Record ID
- UAMeb657627f59b4d39ab61325495213889
- Author
- Journal series
- Commentationes Mathematicae Universitatis Carolinae, ISSN 0010-2628, e-ISSN 1213-7243
- Issue year
- 2008
- Vol
- 49
- Pages
- 633-636
- ASJC Classification
- Language
- en English
- Score (nominal)
- 0
- Score source
- journalList
- Publication indicators
- = 12; : 2012 = 0.148

- Uniform Resource Identifier
- https://researchportal.amu.edu.pl/info/article/UAMeb657627f59b4d39ab61325495213889/

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