7 research outputs found
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
On the Existence of Loose Cycle Tilings and Rainbow Cycles
abstract: Extremal graph theory results often provide minimum degree
conditions which guarantee a copy of one graph exists within
another. A perfect -tiling of a graph is a collection
of subgraphs of such that every element of
is isomorphic to and such that every vertex in
is in exactly one element of . Let denote
the loose cycle on vertices, the -uniform hypergraph
obtained by replacing the edges of a graph cycle
on vertices with edge triples , where is
uniquely assigned to . This dissertation proves for even
, that any sufficiently large -uniform hypergraph
on vertices with minimum -degree
\delta^1(H) \geq {n - 1 \choose 2} - {\Bsize \choose 2} + c(t,n) +
1, where , contains a perfect
-tiling. The result is tight, generalizing previous
results on by Han and Zhao. For an edge colored graph ,
let the minimum color degree be the minimum number of
distinctly colored edges incident to a vertex. Call rainbow if
every edge has a unique color. For , this dissertation
proves that any sufficiently large edge colored graph on
vertices with contains a rainbow
cycle on vertices. The result is tight for odd and
extends previous results for . In addition, for even
, this dissertation proves that any sufficiently large
edge colored graph on vertices with
, where
, contains a rainbow cycle on
vertices. The result is tight when . As a related
result, this dissertation proves for all , that any
sufficiently large oriented graph on vertices with
contains a directed cycle on
vertices. This partially generalizes a result by Kelly,
K\"uhn, and Osthus that uses minimum semidegree rather than minimum
out degree.Dissertation/ThesisDoctoral Dissertation Mathematics 201
Extremal results in hypergraph theory via the absorption method
The so-called "absorbing method" was first introduced in a systematic way by Rödl, Ruciński and Szemerédi in 2006, and has found many uses ever since. Speaking in a general sense, it is useful for finding spanning substructures of combinatorial structures. We establish various results of different natures, in both graph and hypergraph theory, most of them using the absorbing method:
1. We prove an asymptotically best-possible bound on the strong chromatic number with respect to the maximum degree of the graph. This establishes a weak version of a conjecture of Aharoni, Berger and Ziv.
2. We determine asymptotic minimum codegree thresholds which ensure the existence of tilings with tight cycles (of a given size) in uniform hypergraphs. Moreover, we prove results on coverings with tight cycles.
3. We show that every 2-coloured complete graph on the integers contains a monochromatic infinite path whose vertex set is sufficiently "dense" in the natural numbers. This improves results of Galvin and Erdős and of DeBiasio and McKenney