9 research outputs found
Note on bipartite graph tilings
Let s<t be two fixed positive integers. We study what are the minimum degree
conditions for a bipartite graph G, with both color classes of size n=k(s+t),
which ensure that G has a K_{s,t}-factor. Exact result for large n is given.
Our result extends the work of Zhao, who determined the minimum degree
threshold which guarantees that a bipartite graph has a K_{s,s}-factor.Comment: 6 pages, no figures; statement of the main theorem corrected (thanks
to Andrzej Czygrinow and Louis DeBiasio); to appear in SIAM Journal on
Discrete Mathematic
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
Embedding large subgraphs into dense graphs
What conditions ensure that a graph G contains some given spanning subgraph
H? The most famous examples of results of this kind are probably Dirac's
theorem on Hamilton cycles and Tutte's theorem on perfect matchings. Perfect
matchings are generalized by perfect F-packings, where instead of covering all
the vertices of G by disjoint edges, we want to cover G by disjoint copies of a
(small) graph F. It is unlikely that there is a characterization of all graphs
G which contain a perfect F-packing, so as in the case of Dirac's theorem it
makes sense to study conditions on the minimum degree of G which guarantee a
perfect F-packing.
The Regularity lemma of Szemeredi and the Blow-up lemma of Komlos, Sarkozy
and Szemeredi have proved to be powerful tools in attacking such problems and
quite recently, several long-standing problems and conjectures in the area have
been solved using these. In this survey, we give an outline of recent progress
(with our main emphasis on F-packings, Hamiltonicity problems and tree
embeddings) and describe some of the methods involved
Tiling in bipartite graphs with asymmetric minimum degrees
The problem of determining the optimal minimum degree condition for a
balanced bipartite graph on 2ms vertices to contain m vertex disjoint copies of
K_{s,s} was solved by Zhao. Later Hladk\'y and Schacht, and Czygrinow and
DeBiasio determined the optimal minimum degree condition for a balanced
bipartite graph on 2m(s+t) vertices to contain m vertex disjoint copies of
K_{s,t} for fixed positive integers s<t.
For a balanced bipartite graph G[U,V], let \delta_U be the minimum degree
over all vertices in U and \delta_V be the minimum degree over all vertices in
V. We consider the problem of determining the optimal value of
\delta_U+\delta_V which guarantees that G can be tiled with K_{s,s}. We show
that the optimal value depends on D:=|\delta_V-\delta_U|. When D is small, we
show that \delta_U+\delta_V\geq n+3s-5 is best possible. As D becomes larger,
we show that \delta_U+\delta_V can be made smaller, but no smaller than
n+2s-2s^{1/2}. However, when D=n-C for some constant C, we show that there
exist graphs with \delta_U+\delta_V\geq n+s^{s^{1/3}} which cannot be tiled
with K_{s,s}.Comment: 34 pages, 4 figures. This is the unabridged version of the paper,
containing the full proof of Theorem 1.7. The case when |\delta_U-\delta_V|
is small and s>2 involves a lengthy case analysis, spanning pages 20-32; this
section is not included in the "journal version