89 research outputs found
The minimum vertex degree for an almost-spanning tight cycle in a -uniform hypergraph
We prove that any -uniform hypergraph whose minimum vertex degree is at
least admits an almost-spanning
tight cycle, that is, a tight cycle leaving vertices uncovered. The
bound on the vertex degree is asymptotically best possible. Our proof uses the
hypergraph regularity method, and in particular a recent version of the
hypergraph regularity lemma proved by Allen, B\"ottcher, Cooley and Mycroft.Comment: 10 pages. arXiv admin note: text overlap with arXiv:1411.495
Hypergraph matchings and designs
We survey some aspects of the perfect matching problem in hypergraphs, with
particular emphasis on structural characterisation of the existence problem in
dense hypergraphs and the existence of designs.Comment: 19 pages, for the 2018 IC
Covering and tiling hypergraphs with tight cycles
Given , we say that a -uniform hypergraph is a
tight cycle on vertices if there is a cyclic ordering of the vertices of
such that every consecutive vertices under this ordering form an
edge. We prove that if and , then every -uniform
hypergraph on vertices with minimum codegree at least has
the property that every vertex is covered by a copy of . Our result is
asymptotically best possible for infinitely many pairs of and , e.g.
when and are coprime.
A perfect -tiling is a spanning collection of vertex-disjoint copies
of . When is divisible by , the problem of determining the
minimum codegree that guarantees a perfect -tiling was solved by a
result of Mycroft. We prove that if and is not divisible
by and divides , then every -uniform hypergraph on vertices
with minimum codegree at least has a perfect
-tiling. Again our result is asymptotically best possible for infinitely
many pairs of and , e.g. when and are coprime with even.Comment: Revised version, accepted for publication in Combin. Probab. Compu
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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