67 research outputs found
Packing k-partite k-uniform hypergraphs
Let and be -graphs (-uniform hypergraphs); then a perfect
-packing in is a collection of vertex-disjoint copies of in
which together cover every vertex of . For any fixed let
be the minimum such that any -graph on vertices with
minimum codegree contains a perfect -packing. The
problem of determining has been widely studied for graphs (i.e.
-graphs), but little is known for . Here we determine the
asymptotic value of for all complete -partite -graphs ,
as well as a wide class of other -partite -graphs. In particular, these
results provide an asymptotic solution to a question of R\"odl and Ruci\'nski
on the value of when is a loose cycle. We also determine
asymptotically the codegree threshold needed to guarantee an -packing
covering all but a constant number of vertices of for any complete
-partite -graph .Comment: v2: Updated with minor corrections. Accepted for publication in
Journal of Combinatorial Theory, Series
A Geometric Theory for Hypergraph Matching
We develop a theory for the existence of perfect matchings in hypergraphs
under quite general conditions. Informally speaking, the obstructions to
perfect matchings are geometric, and are of two distinct types: 'space
barriers' from convex geometry, and 'divisibility barriers' from arithmetic
lattice-based constructions. To formulate precise results, we introduce the
setting of simplicial complexes with minimum degree sequences, which is a
generalisation of the usual minimum degree condition. We determine the
essentially best possible minimum degree sequence for finding an almost perfect
matching. Furthermore, our main result establishes the stability property:
under the same degree assumption, if there is no perfect matching then there
must be a space or divisibility barrier. This allows the use of the stability
method in proving exact results. Besides recovering previous results, we apply
our theory to the solution of two open problems on hypergraph packings: the
minimum degree threshold for packing tetrahedra in 3-graphs, and Fischer's
conjecture on a multipartite form of the Hajnal-Szemer\'edi Theorem. Here we
prove the exact result for tetrahedra and the asymptotic result for Fischer's
conjecture; since the exact result for the latter is technical we defer it to a
subsequent paper.Comment: Accepted for publication in Memoirs of the American Mathematical
Society. 101 pages. v2: minor changes including some additional diagrams and
passages of expository tex
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
Classification of maximum hittings by large families
For integers and , where is sufficiently large, and for every set
we determine the maximal left-compressed intersecting
families which achieve maximum hitting
with (i.e. have the most members which intersect ). This answers a
question of Barber, who extended previous results by Borg to characterise those
sets for which maximum hitting is achieved by the star.Comment: v2: minor corrections in response to reviewer comments. To appear in
Graphs and Combinatoric
Hamilton cycles in hypergraphs below the Dirac threshold
We establish a precise characterisation of -uniform hypergraphs with
minimum codegree close to which contain a Hamilton -cycle. As an
immediate corollary we identify the exact Dirac threshold for Hamilton
-cycles in -uniform hypergraphs. Moreover, by derandomising the proof of
our characterisation we provide a polynomial-time algorithm which, given a
-uniform hypergraph with minimum codegree close to , either finds a
Hamilton -cycle in or provides a certificate that no such cycle exists.
This surprising result stands in contrast to the graph setting, in which below
the Dirac threshold it is NP-hard to determine if a graph is Hamiltonian. We
also consider tight Hamilton cycles in -uniform hypergraphs for , giving a series of reductions to show that it is NP-hard to determine
whether a -uniform hypergraph with minimum degree contains a tight Hamilton cycle. It is therefore
unlikely that a similar characterisation can be obtained for tight Hamilton
cycles.Comment: v2: minor revisions in response to reviewer comments, most pseudocode
and details of the polynomial time reduction moved to the appendix which will
not appear in the printed version of the paper. To appear in Journal of
Combinatorial Theory, Series
Polynomial-time perfect matchings in dense hypergraphs
Let be a -graph on vertices, with minimum codegree at least for some fixed . In this paper we construct a polynomial-time
algorithm which finds either a perfect matching in or a certificate that
none exists. This essentially solves a problem of Karpi\'nski, Ruci\'nski and
Szyma\'nska; Szyma\'nska previously showed that this problem is NP-hard for a
minimum codegree of . Our algorithm relies on a theoretical result of
independent interest, in which we characterise any such hypergraph with no
perfect matching using a family of lattice-based constructions.Comment: 64 pages. Update includes minor revisions. To appear in Advances in
Mathematic
A random version of Sperner's theorem
Let denote the power set of , ordered by inclusion, and
let be obtained from by selecting elements
from independently at random with probability . A classical
result of Sperner asserts that every antichain in has size at
most that of the middle layer, . In this note
we prove an analogous result for : If then, with high probability, the size of the largest antichain in
is at most . This
solves a conjecture of Osthus who proved the result in the case when . Our condition on is best-possible. In fact, we prove a
more general result giving an upper bound on the size of the largest antichain
for a wider range of values of .Comment: 7 pages. Updated to include minor revisions and publication dat
Hamilton cycles in quasirandom hypergraphs
We show that, for a natural notion of quasirandomness in -uniform
hypergraphs, any quasirandom -uniform hypergraph on vertices with
constant edge density and minimum vertex degree contains a
loose Hamilton cycle. We also give a construction to show that a -uniform
hypergraph satisfying these conditions need not contain a Hamilton -cycle
if divides . The remaining values of form an interesting
open question.Comment: 18 pages. Accepted for publication in Random Structures & Algorithm
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