Perfect matchings in large uniform hypergraphs with large minimum collective degree

  • Authors:

  • Vojtech Rödl;Andrzej Ruciński;Endre Szemerédi

  • Affiliations:

  • Emory University, Atlanta, GA, USA;A. Mickiewicz University, Poznań, Poland;Rutgers University, New Brunswick, NJ, USA

  • Venue:
  • Journal of Combinatorial Theory Series A
  • Year:
  • 2009

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Abstract

We define a perfect matching in a k-uniform hypergraph H on n vertices as a set of @?n/k@? disjoint edges. Let @d"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. In this paper we study the relation between @d"k"-"1(H) and the presence of a perfect matching in H for k=3. Let t(k,n) be the smallest integer t such that every k-uniform hypergraph on n vertices and with @d"k"-"1(H)=t contains a perfect matching. For large n divisible by k, we completely determine the values of t(k,n), which turn out to be very close to n/2-k. For example, if k is odd and n is large and even, then t(k,n)=n/2-k+2. In contrast, for n not divisible by k, we show that t(k,n)~n/k. In the proofs we employ a newly developed ''absorbing'' technique, which has a potential to be applicable in a more general context of establishing existence of spanning subgraphs of graphs and hypergraphs.