1,622 research outputs found
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
On globally sparse Ramsey graphs
We say that a graph has the Ramsey property w.r.t.\ some graph and
some integer , or is -Ramsey for short, if any -coloring
of the edges of contains a monochromatic copy of . R{\"o}dl and
Ruci{\'n}ski asked how globally sparse -Ramsey graphs can possibly
be, where the density of is measured by the subgraph with
the highest average degree. So far, this so-called Ramsey density is known only
for cliques and some trivial graphs . In this work we determine the Ramsey
density up to some small error terms for several cases when is a complete
bipartite graph, a cycle or a path, and colors are available
On-line Ramsey numbers
Consider the following game between two players, Builder and Painter. Builder
draws edges one at a time and Painter colours them, in either red or blue, as
each appears. Builder's aim is to force Painter to draw a monochromatic copy of
a fixed graph G. The minimum number of edges which Builder must draw,
regardless of Painter's strategy, in order to guarantee that this happens is
known as the on-line Ramsey number \tilde{r}(G) of G. Our main result, relating
to the conjecture that \tilde{r}(K_t) = o(\binom{r(t)}{2}), is that there
exists a constant c > 1 such that \tilde{r}(K_t) \leq c^{-t} \binom{r(t)}{2}
for infinitely many values of t. We also prove a more specific upper bound for
this number, showing that there exists a constant c such that \tilde{r}(K_t)
\leq t^{-c \frac{\log t}{\log \log t}} 4^t. Finally, we prove a new upper bound
for the on-line Ramsey number of the complete bipartite graph K_{t,t}.Comment: 11 page
Density theorems for bipartite graphs and related Ramsey-type results
In this paper, we present several density-type theorems which show how to
find a copy of a sparse bipartite graph in a graph of positive density. Our
results imply several new bounds for classical problems in graph Ramsey theory
and improve and generalize earlier results of various researchers. The proofs
combine probabilistic arguments with some combinatorial ideas. In addition,
these techniques can be used to study properties of graphs with a forbidden
induced subgraph, edge intersection patterns in topological graphs, and to
obtain several other Ramsey-type statements
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