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Nearly perfect matchings in regular simple hypergraphs Jeong-Han Kim
 

Summary: Nearly perfect matchings in regular simple hypergraphs
Noga Alon
Jeong-Han Kim
Joel Spencer
Abstract
For every fixed k 3 there exists a constant ck with the following property. Let H be a k-
uniform, D-regular hypergraph on N vertices, in which no two edges contain more than one com-
mon vertex. If k > 3 then H contains a matching covering all vertices but at most ckND-1/(k-1)
.
If k = 3, then H contains a matching covering all vertices but at most c3ND-1/2
ln3/2
D. This
improves previous estimates and implies, for example, that any Steiner Triple System on N ver-
tices contains a matching covering all vertices but at most O(N1/2
ln3/2
N), improving results by
various authors.
1 Introduction
A Hypergraph is a pair (V, H), where V is a finite set of vertices and H is a finite family of subsets
of V , called edges. It is k-uniform if every edge contains precisely k vertices. The degree deg(x) is

  

Source: Alon, Noga - School of Mathematical Sciences, Tel Aviv University

 

Collections: Mathematics