446 research outputs found
Matchings and Tilings in Hypergraphs
We consider two extremal problems in hypergraphs. First, given k ≥ 3 and k-partite k-uniform hypergraphs, as a generalization of graph (k = 2) matchings, we determine the partite minimum codegree threshold for matchings with at most one vertex left in each part, thereby answering a problem asked by R ̈odl and Rucin ́ski. We further improve the partite minimum codegree conditions to sum of all k partite codegrees, in which case the partite minimum codegree is not necessary large.
Second, as a generalization of (hyper)graph matchings, we determine the minimum vertex degree threshold asymptotically for perfect Ka,b,c-tlings in large 3-uniform hypergraphs, where Ka,b,c is any complete 3-partite 3-uniform hypergraphs with each part of size a, b and c. This partially answers a question of Mycroft, who proved an analogous result with respect to codegree for r-uniform hypergraphs for all r ≥ 3. Our proof uses Regularity Lemma, the absorbing method, fractional tiling, and a recent result on shadows for 3-graphs
Rainbow perfect matchings in r-partite graph structures
A matching M in an edge–colored (hyper)graph is rainbow if each pair of edges in M have distinct colors. We extend the result of Erdos and Spencer on the existence of rainbow perfect matchings in the complete bipartite graph Kn,n to complete bipartite multigraphs, dense regular bipartite graphs and complete r-partite r-uniform hypergraphs. The proof of the results use the Lopsided version of the Local Lovász Lemma.Peer ReviewedPostprint (author's final draft
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 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
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
- …