225 research outputs found
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
The approximate Loebl-Koml\'os-S\'os Conjecture II: The rough structure of LKS graphs
This is the second of a series of four papers in which we prove the following
relaxation of the Loebl-Komlos--Sos Conjecture: For every there
exists a number such that for every every -vertex graph
with at least vertices of degree at least
contains each tree of order as a subgraph.
In the first paper of the series, we gave a decomposition of the graph
into several parts of different characteristics; this decomposition might be
viewed as an analogue of a regular partition for sparse graphs. In the present
paper, we find a combinatorial structure inside this decomposition. In the last
two papers, we refine the structure and use it for embedding the tree .Comment: 38 pages, 4 figures; new is Section 5.1.1; accepted to SIDM
Polychromatic Coloring for Half-Planes
We prove that for every integer , every finite set of points in the plane
can be -colored so that every half-plane that contains at least
points, also contains at least one point from every color class. We also show
that the bound is best possible. This improves the best previously known
lower and upper bounds of and respectively. We also show
that every finite set of half-planes can be colored so that if a point
belongs to a subset of at least of the half-planes then
contains a half-plane from every color class. This improves the best previously
known upper bound of . Another corollary of our first result is a new
proof of the existence of small size \eps-nets for points in the plane with
respect to half-planes.Comment: 11 pages, 5 figure
The Approximate Loebl-Koml\'os-S\'os Conjecture III: The finer structure of LKS graphs
This is the third of a series of four papers in which we prove the following
relaxation of the Loebl-Komlos-Sos Conjecture: For every there
exists a number such that for every every -vertex graph
with at least vertices of degree at least
contains each tree of order as a subgraph.
In the first paper of the series, we gave a decomposition of the graph
into several parts of different characteristics. In the second paper, we found
a combinatorial structure inside the decomposition. In this paper, we will give
a refinement of this structure. In the forthcoming fourth paper, the refined
structure will be used for embedding the tree .Comment: 59 pages, 4 figures; further comments by a referee incorporated; this
includes a subtle but important fix to Lemma 5.1; as a consequence,
Preconfiguration Clubs was change
The approximate Loebl-Koml\'os-S\'os Conjecture IV: Embedding techniques and the proof of the main result
This is the last paper of a series of four papers in which we prove the
following relaxation of the Loebl-Komlos-Sos Conjecture: For every
there exists a number~ such that for every every -vertex graph
with at least vertices of degree at least
contains each tree of order as a subgraph.
In the first two papers of this series, we decomposed the host graph , and
found a suitable combinatorial structure inside the decomposition. In the third
paper, we refined this structure, and proved that any graph satisfying the
conditions of the above approximate version of the Loebl-Komlos-Sos Conjecture
contains one of ten specific configurations. In this paper we embed the tree
in each of the ten configurations.Comment: 81 pages, 12 figures. A fix reflecting the change of Preconfiguration
Clubs in Paper III, additional small change
Powers of Hamilton cycles in pseudorandom graphs
We study the appearance of powers of Hamilton cycles in pseudorandom graphs,
using the following comparatively weak pseudorandomness notion. A graph is
-pseudorandom if for all disjoint and with and we have
. We prove that for all there is an
such that an -pseudorandom graph on
vertices with minimum degree at least contains the square of a
Hamilton cycle. In particular, this implies that -graphs with
contain the square of a Hamilton cycle, and thus
a triangle factor if is a multiple of . This improves on a result of
Krivelevich, Sudakov and Szab\'o [Triangle factors in sparse pseudo-random
graphs, Combinatorica 24 (2004), no. 3, 403--426].
We also extend our result to higher powers of Hamilton cycles and establish
corresponding counting versions.Comment: 30 pages, 1 figur
Convergence in measure under Finite Additivity
We investigate the possibility of replacing the topology of convergence in
probability with convergence in . A characterization of continuous linear
functionals on the space of measurable functions is also obtained
The approximate Loebl-Komlós-Sós conjecture I: The sparse decomposition
In a series of four papers we prove the following relaxation of the Loebl-Komlós-Sós conjecture: For every α > 0 there exists a number k0 such that for every k > k0, every n-vertex graph G with at least (1/2 + α)n vertices of degree at least (1 + α)k contains each tree T of order k as a subgraph. The method to prove our result follows a strategy similar to approaches that employ the Szemerédi regularity lemma: We decompose the graph G, find a suitable combinatorial structure inside the decomposition, and then embed the tree T into G using this structure. Since for sparse graphs G, the decomposition given by the regularity lemma is not helpful, we use a more general decomposition technique. We show that each graph can be decomposed into vertices of huge degree, regular pairs (in the sense of the regularity lemma), and two other objects each exhibiting certain expansion properties. In this paper, we introduce this novel decomposition technique. In the three follow-up papers, we find a suitable combinatorial structure inside the decomposition, which we then use for embedding the tree. © 2017 the authors
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