227 research outputs found
A note on perfect matchings in uniform hypergraphs
We determine the \emph{exact} minimum -degree threshold for perfect
matchings in -uniform hypergraphs when the corresponding threshold for
perfect fractional matchings is significantly less than . This extends our previous results that determine the
minimum -degree thresholds for perfect matchings in -uniform
hypergraphs for all and provides two new (exact) thresholds:
and .Comment: 11 pages, final versio
Exact minimum degree thresholds for perfect matchings in uniform hypergraphs II
Given positive integers k\geq 3 and r where k/2 \leq r \leq k-1, we give a
minimum r-degree condition that ensures a perfect matching in a k-uniform
hypergraph. This condition is best possible and improves on work of Pikhurko
who gave an asymptotically exact result. Our approach makes use of the
absorbing method, and builds on work in 'Exact minimum degree thresholds for
perfect matchings in uniform hypergraphs', where we proved the result for k
divisible by 4.Comment: 20 pages, 1 figur
Large matchings in uniform hypergraphs and the conjectures of Erdos and Samuels
In this paper we study conditions which guarantee the existence of perfect
matchings and perfect fractional matchings in uniform hypergraphs. We reduce
this problem to an old conjecture by Erd\H{o}s 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
-degree ensuring the existence of perfect (fractional) matchings. In
particular, we asymptotically determine 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
Matchings in 3-uniform hypergraphs
We determine the minimum vertex degree that ensures a perfect matching in a
3-uniform hypergraph. More precisely, suppose that H is a sufficiently large
3-uniform hypergraph whose order n is divisible by 3. If the minimum vertex
degree of H is greater than \binom{n-1}{2}-\binom{2n/3}{2}, then H contains a
perfect matching. This bound is tight and answers a question of Han, Person and
Schacht. More generally, we show that H contains a matching of size d\le n/3 if
its minimum vertex degree is greater than \binom{n-1}{2}-\binom{n-d}{2}, which
is also best possible. This extends a result of Bollobas, Daykin and Erdos.Comment: 18 pages, 1 figure. To appear in JCT
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