371 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
The history of degenerate (bipartite) extremal graph problems
This paper is a survey on Extremal Graph Theory, primarily focusing on the
case when one of the excluded graphs is bipartite. On one hand we give an
introduction to this field and also describe many important results, methods,
problems, and constructions.Comment: 97 pages, 11 figures, many problems. This is the preliminary version
of our survey presented in Erdos 100. In this version 2 only a citation was
complete
Extremal results in sparse pseudorandom graphs
Szemer\'edi's regularity lemma is a fundamental tool in extremal
combinatorics. However, the original version is only helpful in studying dense
graphs. In the 1990s, Kohayakawa and R\"odl proved an analogue of Szemer\'edi's
regularity lemma for sparse graphs as part of a general program toward
extending extremal results to sparse graphs. Many of the key applications of
Szemer\'edi's regularity lemma use an associated counting lemma. In order to
prove extensions of these results which also apply to sparse graphs, it
remained a well-known open problem to prove a counting lemma in sparse graphs.
The main advance of this paper lies in a new counting lemma, proved following
the functional approach of Gowers, which complements the sparse regularity
lemma of Kohayakawa and R\"odl, allowing us to count small graphs in regular
subgraphs of a sufficiently pseudorandom graph. We use this to prove sparse
extensions of several well-known combinatorial theorems, including the removal
lemmas for graphs and groups, the Erd\H{o}s-Stone-Simonovits theorem and
Ramsey's theorem. These results extend and improve upon a substantial body of
previous work.Comment: 70 pages, accepted for publication in Adv. Mat
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