94 research outputs found
Asymptotic multipartite version of the Alon-Yuster theorem
In this paper, we prove the asymptotic multipartite version of the
Alon-Yuster theorem, which is a generalization of the Hajnal-Szemer\'edi
theorem: If is an integer, is a -colorable graph and
is fixed, then, for every sufficiently large , where
divides , and for every balanced -partite graph on vertices with
each of its corresponding bipartite subgraphs having minimum
degree at least , has a subgraph consisting of
vertex-disjoint copies of .
The proof uses the Regularity method together with linear programming.Comment: 22 pages, 1 figur
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
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
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