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Large matchings in uniform hypergraphs and the conjectures of Erdos and Samuels

Summary: Large matchings in uniform hypergraphs and the conjectures of
Erdos and Samuels
Noga Alon
Peter Frankl
Hao Huang
Vojtech R¨odl §
Andrzej Ruci´nski ¶
Benny Sudakov
In this paper we study conditions which guarantee the existence of perfect matchings and per-
fect fractional matchings in uniform hypergraphs. We reduce this problem to an old conjecture
by Erdos on estimating the maximum number of edges in a hypergraph when the (fractional)
matching number is given, which we are able to solve in some special cases using probabilistic
techniques. Based on these results, we obtain some general theorems on the minimum d-degree
ensuring the existence of perfect (fractional) matchings. In particular, we asymptotically deter-
mine the minimum vertex degree which guarantees a perfect matching in 4-uniform and 5-uniform
hypergraphs. We also discuss an application to a problem of finding an optimal data allocation
in a distributed storage system.
1 Introduction
A k-uniform hypergraph or a k-graph for short, is a pair H = (V, E), where V := V (H) is a finite


Source: Alon, Noga - School of Mathematical Sciences, Tel Aviv University
Ruciński, Andrzej - Department of Discrete Mathematics, Adam Mickiewicz University
Sudakov, Benjamin - Department of Mathematics, University of California at Los Angeles


Collections: Mathematics