295 research outputs found
Countable connected-homogeneous digraphs
A digraph is connected-homogeneous if every isomorphism between two finite
connected induced subdigraphs extends to an automorphism of the whole digraph.
In this paper, we completely classify the countable connected-homogeneous
digraphs.Comment: 49 page
Homogeneous 2-partite digraphs
We call a 2-partite digraph D homogeneous if every isomorphism between finite
induced subdigraphs that respects the 2-partition of D extends to an
automorphism of D that does the same. In this note, we classify the homogeneous
2-partite digraphs.Comment: 5 page
Hamilton cycles in graphs and hypergraphs: an extremal perspective
As one of the most fundamental and well-known NP-complete problems, the
Hamilton cycle problem has been the subject of intensive research. Recent
developments in the area have highlighted the crucial role played by the
notions of expansion and quasi-randomness. These concepts and other recent
techniques have led to the solution of several long-standing problems in the
area. New aspects have also emerged, such as resilience, robustness and the
study of Hamilton cycles in hypergraphs. We survey these developments and
highlight open problems, with an emphasis on extremal and probabilistic
approaches.Comment: to appear in the Proceedings of the ICM 2014; due to given page
limits, this final version is slightly shorter than the previous arxiv
versio
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
Tournaments, 4-uniform hypergraphs, and an exact extremal result
We consider -uniform hypergraphs with the maximum number of hyperedges
subject to the condition that every set of vertices spans either or
exactly hyperedges and give a construction, using quadratic residues, for
an infinite family of such hypergraphs with the maximum number of hyperedges.
Baber has previously given an asymptotically best-possible result using random
tournaments. We give a connection between Baber's result and our construction
via Paley tournaments and investigate a `switching' operation on tournaments
that preserves hypergraphs arising from this construction.Comment: 23 pages, 6 figure
Large unavoidable subtournaments
Let denote the tournament on vertices consisting of three disjoint
vertex classes and of size , each of which is oriented as a
transitive subtournament, and with edges directed from to , from
to and from to . Fox and Sudakov proved that given a
natural number and there is such that
every tournament of order which is -far from
being transitive contains as a subtournament. Their proof showed that
and they conjectured that
this could be reduced to . Here we
prove this conjecture.Comment: 9 page
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