4,624 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 small Mixed Pattern Ramsey numbers
We call the minimum order of any complete graph so that for any coloring of
the edges by colors it is impossible to avoid a monochromatic or rainbow
triangle, a Mixed Ramsey number. For any graph with edges colored from the
above set of colors, if we consider the condition of excluding in the
above definition, we produce a \emph{Mixed Pattern Ramsey number}, denoted
. We determine this function in terms of for all colored -cycles
and all colored -cliques. We also find bounds for when is a
monochromatic odd cycles, or a star for sufficiently large . We state
several open questions.Comment: 16 page
Cycles are strongly Ramsey-unsaturated
We call a graph H Ramsey-unsaturated if there is an edge in the complement of
H such that the Ramsey number r(H) of H does not change upon adding it to H.
This notion was introduced by Balister, Lehel and Schelp who also proved that
cycles (except for ) are Ramsey-unsaturated, and conjectured that,
moreover, one may add any chord without changing the Ramsey number of the cycle
, unless n is even and adding the chord creates an odd cycle.
We prove this conjecture for large cycles by showing a stronger statement: If
a graph H is obtained by adding a linear number of chords to a cycle ,
then , as long as the maximum degree of H is bounded, H is either
bipartite (for even n) or almost bipartite (for odd n), and n is large.
This motivates us to call cycles strongly Ramsey-unsaturated. Our proof uses
the regularity method
On path-quasar Ramsey numbers
Let and be two given graphs. The Ramsey number is
the least integer such that for every graph on vertices, either
contains a or contains a . Parsons gave a recursive
formula to determine the values of , where is a path on
vertices and is a star on vertices. In this note, we first
give an explicit formula for the path-star Ramsey numbers. Secondly, we study
the Ramsey numbers , where is a linear forest on
vertices. We determine the exact values of for the cases
and , and for the case that has no odd component.
Moreover, we give a lower bound and an upper bound for the case and has at least one odd component.Comment: 7 page
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