1,044 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
From Super-Yang-Mills Theory to QCD: Planar Equivalence and its Implications
We review and extend our recent work on the planar (large N) equivalence
between gauge theories with varying degree of supersymmetry. The main emphasis
is made on the planar equivalence between N=1 gluodynamics (super-Yang-Mills
theory) and a non-supersymmetric "orientifold field theory." We outline an
"orientifold" large N expansion, analyze its possible phenomenological
consequences in one-flavor massless QCD, and make a first attempt at extending
the correspondence to three massless flavors. An analytic calculation of the
quark condensate in one-flavor QCD starting from the gluino condensate in N=1
gluodynamics is thoroughly discussed. We also comment on a planar equivalence
involving N=2 supersymmetry, on "chiral rings" in non-supersymmetric theories,
and on the origin of planar equivalence from an underlying, non-tachyonic
type-0 string theory. Finally, possible further directions of investigation,
such as the gauge/gravity correspondence in large-N orientifold field theory,
are briefly discussed.Comment: 106 pages, LaTex. 15 figures. v2:minor changes, refs. added. To be
published in the Ian Kogan Memorial Collection "From Fields to Strings:
Circumnavigating Theoretical Physics," World Scientific, 200
Towards a constrained Willmore conjecture
We give an overview of the constrained Willmore problem and address some
conjectures arising from partial results and numerical experiments.
Ramifications of these conjectures would lead to a deeper understanding of the
Willmore functional over conformal immersions from compact surfaces.Comment: 17page
Problems in extremal graph theory
We consider a variety of problems in extremal graph and set theory.
The {\em chromatic number} of , , is the smallest integer
such that is -colorable.
The {\it square} of , written , is the supergraph of in which also
vertices within distance 2 of each other in are adjacent.
A graph is a {\it minor} of if
can be obtained from a subgraph of by contracting edges.
We show that the upper bound for
conjectured by Wegner (1977) for planar graphs
holds when is a -minor-free graph.
We also show that is equal to the bound
only when contains a complete graph of that order.
One of the central problems of extremal hypergraph theory is
finding the maximum number of edges in a hypergraph
that does not contain a specific forbidden structure.
We consider as a forbidden structure a fixed number of members
that have empty common intersection
as well as small union.
We obtain a sharp upper bound on the size of uniform hypergraphs
that do not contain this structure,
when the number of vertices is sufficiently large.
Our result is strong enough to imply the same sharp upper bound
for several other interesting forbidden structures
such as the so-called strong simplices and clusters.
The {\em -dimensional hypercube}, ,
is the graph whose vertex set is and
whose edge set consists of the vertex pairs
differing in exactly one coordinate.
The generalized Tur\'an problem asks for the maximum number
of edges in a subgraph of a graph that does not contain
a forbidden subgraph .
We consider the Tur\'an problem where is and
is a cycle of length with .
Confirming a conjecture of Erd{\H o}s (1984),
we show that the ratio of the size of such a subgraph of
over the number of edges of is ,
i.e. in the limit this ratio approaches 0
as approaches infinity
Recommended from our members
Combinatorics
This is the report on the Oberwolfach workshop on Combinatorics, held 1–7 January 2006. Combinatorics is a branch of mathematics studying families of mainly, but not exclusively, finite or countable structures – discrete objects. The discrete objects considered in the workshop were graphs, set systems, discrete geometries, and matrices. The programme consisted of 15 invited lectures, 18 contributed talks, and a problem session focusing on recent developments in graph theory, coding theory, discrete geometry, extremal combinatorics, Ramsey theory, theoretical computer science, and probabilistic combinatorics
K_6 minors in 6-connected graphs of bounded tree-width
We prove that every sufficiently big 6-connected graph of bounded tree-width
either has a K_6 minor, or has a vertex whose deletion makes the graph planar.
This is a step toward proving that the same conclusion holds for all
sufficiently big 6-connected graphs. Jorgensen conjectured that it holds for
all 6-connected graphs.Comment: 33 pages, 8 figure
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