40 research outputs found
On the bipartite graph packing problem
The graph packing problem is a well-known area in graph theory. We consider a
bipartite version and give almost tight conditions on the packability of two
bipartite sequences
On embedding well-separable graphs
Call a simple graph of order well-separable, if by deleting a
separator set of size the leftover will have components of size at most
. We prove, that bounded degree well-separable spanning subgraphs are
easy to embed: for every and positive integer there exists
an such that if , for a well-separable
graph of order and
for a simple graph of order , then . We extend our result
to graphs with small band-width, too.Comment: 11 pages, submitted for publicatio
Spanning embeddings of arrangeable graphs with sublinear bandwidth
The Bandwidth Theorem of B\"ottcher, Schacht and Taraz [Mathematische Annalen
343 (1), 175-205] gives minimum degree conditions for the containment of
spanning graphs H with small bandwidth and bounded maximum degree. We
generalise this result to a-arrangeable graphs H with \Delta(H)<sqrt(n)/log(n),
where n is the number of vertices of H.
Our result implies that sufficiently large n-vertex graphs G with minimum
degree at least (3/4+\gamma)n contain almost all planar graphs on n vertices as
subgraphs. Using techniques developed by Allen, Brightwell and Skokan
[Combinatorica, to appear] we can also apply our methods to show that almost
all planar graphs H have Ramsey number at most 12|H|. We obtain corresponding
results for graphs embeddable on different orientable surfaces.Comment: 20 page
Spanning Trees in Graphs of High Minimum Degree which have a Universal Vertex II: A Tight Result
We prove that, if is sufficiently large, every graph on vertices
that has a universal vertex and minimum degree at least contains each tree with edges as a subgraph. Our result
confirms, for large , an important special case of a conjecture by Havet,
Reed, Stein, and Wood.
The present paper builds on the results of a companion paper in which we
proved the statement for all trees having a vertex that is adjacent to many
leaves.Comment: 29 page
Transversal factors and spanning trees
Given a collection of graphs with the same
vertex set, an -edge graph is a transversal if
there is a bijection such that for
each . We give asymptotically-tight minimum degree conditions for a
graph collection on an -vertex set to have a transversal which is a copy of
a graph , when is an -vertex graph which is an -factor or a tree
with maximum degree .Comment: 21 page
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
Spanning Trees in Graphs of High Minimum Degree with a Universal Vertex I: An Approximate Asymptotic Result
In this paper and a companion paper, we prove that, if is sufficiently
large, every graph on vertices that has a universal vertex and minimum
degree at least contains each tree with
edges as a subgraph. The present paper already contains an approximate
asymptotic version of the result.
Our result confirms, for large , an important special case of a recent
conjecture by Havet, Reed, Stein, and Wood.Comment: 46 page