253 research outputs found
Bandwidth theorem for random graphs
A graph is said to have \textit{bandwidth} at most , if there exists a
labeling of the vertices by , so that whenever
is an edge of . Recently, B\"{o}ttcher, Schacht, and Taraz
verified a conjecture of Bollob\'{a}s and Koml\'{o}s which says that for every
positive , there exists such that if is an
-vertex -chromatic graph with maximum degree at most which has
bandwidth at most , then any graph on vertices with minimum
degree at least contains a copy of for large enough
. In this paper, we extend this theorem to dense random graphs. For
bipartite , this answers an open question of B\"{o}ttcher, Kohayakawa, and
Taraz. It appears that for non-bipartite the direct extension is not
possible, and one needs in addition that some vertices of have independent
neighborhoods. We also obtain an asymptotically tight bound for the maximum
number of vertex disjoint copies of a fixed -chromatic graph which one
can find in a spanning subgraph of with minimum degree .Comment: 29 pages, 3 figure
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 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
A polyhedral approach for the Equitable Coloring Problem
In this work we study the polytope associated with a 0,1-integer programming
formulation for the Equitable Coloring Problem. We find several families of
valid inequalities and derive sufficient conditions in order to be
facet-defining inequalities. We also present computational evidence that shows
the efficacy of these inequalities used in a cutting-plane algorithm
Supersaturation Problem for Color-Critical Graphs
The \emph{Tur\'an function} \ex(n,F) of a graph is the maximum number
of edges in an -free graph with vertices. The classical results of
Tur\'an and Rademacher from 1941 led to the study of supersaturated graphs
where the key question is to determine , the minimum number of copies
of that a graph with vertices and \ex(n,F)+q edges can have.
We determine asymptotically when is \emph{color-critical}
(that is, contains an edge whose deletion reduces its chromatic number) and
.
Determining the exact value of seems rather difficult. For
example, let be the limit superior of for which the extremal
structures are obtained by adding some edges to a maximum -free graph.
The problem of determining for cliques was a well-known question of Erd\H
os that was solved only decades later by Lov\'asz and Simonovits. Here we prove
that for every {color-critical}~. Our approach also allows us to
determine for a number of graphs, including odd cycles, cliques with one
edge removed, and complete bipartite graphs plus an edge.Comment: 27 pages, 2 figure
An Ore-type theorem for perfect packings in graphs
We say that a graph G has a perfect H-packing (also called an H-factor) if
there exists a set of disjoint copies of H in G which together cover all the
vertices of G. Given a graph H, we determine, asymptotically, the Ore-type
degree condition which ensures that a graph G has a perfect H-packing. More
precisely, let \delta_{\rm Ore} (H,n) be the smallest number k such that every
graph G whose order n is divisible by |H| and with d(x)+d(y)\geq k for all
non-adjacent x \not = y \in V(G) contains a perfect H-packing. We determine
\lim_{n\to \infty} \delta_{\rm Ore} (H,n)/n.Comment: 23 pages, 1 figure. Extra examples and a sketch proof of Theorem 4
added. To appear in the SIAM Journal on Discrete Mathematic
On the Chromatic Thresholds of Hypergraphs
Let F be a family of r-uniform hypergraphs. The chromatic threshold of F is
the infimum of all non-negative reals c such that the subfamily of F comprising
hypergraphs H with minimum degree at least has bounded
chromatic number. This parameter has a long history for graphs (r=2), and in
this paper we begin its systematic study for hypergraphs.
{\L}uczak and Thomass\'e recently proved that the chromatic threshold of the
so-called near bipartite graphs is zero, and our main contribution is to
generalize this result to r-uniform hypergraphs. For this class of hypergraphs,
we also show that the exact Tur\'an number is achieved uniquely by the complete
(r+1)-partite hypergraph with nearly equal part sizes. This is one of very few
infinite families of nondegenerate hypergraphs whose Tur\'an number is
determined exactly. In an attempt to generalize Thomassen's result that the
chromatic threshold of triangle-free graphs is 1/3, we prove bounds for the
chromatic threshold of the family of 3-uniform hypergraphs not containing {abc,
abd, cde}, the so-called generalized triangle.
In order to prove upper bounds we introduce the concept of fiber bundles,
which can be thought of as a hypergraph analogue of directed graphs. This leads
to the notion of fiber bundle dimension, a structural property of fiber bundles
that is based on the idea of Vapnik-Chervonenkis dimension in hypergraphs. Our
lower bounds follow from explicit constructions, many of which use a hypergraph
analogue of the Kneser graph. Using methods from extremal set theory, we prove
that these Kneser hypergraphs have unbounded chromatic number. This generalizes
a result of Szemer\'edi for graphs and might be of independent interest. Many
open problems remain.Comment: 37 pages, 4 figure
Edge-decompositions of graphs with high minimum degree
A fundamental theorem of Wilson states that, for every graph , every
sufficiently large -divisible clique has an -decomposition. Here a graph
is -divisible if divides and the greatest common divisor
of the degrees of divides the greatest common divisor of the degrees of
, and has an -decomposition if the edges of can be covered by
edge-disjoint copies of . We extend this result to graphs which are
allowed to be far from complete. In particular, together with a result of
Dross, our results imply that every sufficiently large -divisible graph of
minimum degree at least has a -decomposition. This
significantly improves previous results towards the long-standing conjecture of
Nash-Williams that every sufficiently large -divisible graph with minimum
degree at least has a -decomposition. We also obtain the
asymptotically correct minimum degree thresholds of for the
existence of a -decomposition, and of for the existence of a
-decomposition, where . Our main contribution is a
general `iterative absorption' method which turns an approximate or fractional
decomposition into an exact one. In particular, our results imply that in order
to prove an asymptotic version of Nash-Williams' conjecture, it suffices to
show that every -divisible graph with minimum degree at least
has an approximate -decomposition,Comment: 41 pages. This version includes some minor corrections, updates and
improvement
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