406 research outputs found
Combinatorial theorems relative to a random set
We describe recent advances in the study of random analogues of combinatorial
theorems.Comment: 26 pages. Submitted to Proceedings of the ICM 201
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
Ramsey numbers of ordered graphs
An ordered graph is a pair where is a graph and
is a total ordering of its vertices. The ordered Ramsey number
is the minimum number such that every ordered
complete graph with vertices and with edges colored by two colors contains
a monochromatic copy of .
In contrast with the case of unordered graphs, we show that there are
arbitrarily large ordered matchings on vertices for which
is superpolynomial in . This implies that
ordered Ramsey numbers of the same graph can grow superpolynomially in the size
of the graph in one ordering and remain linear in another ordering.
We also prove that the ordered Ramsey number is
polynomial in the number of vertices of if the bandwidth of
is constant or if is an ordered graph of constant
degeneracy and constant interval chromatic number. The first result gives a
positive answer to a question of Conlon, Fox, Lee, and Sudakov.
For a few special classes of ordered paths, stars or matchings, we give
asymptotically tight bounds on their ordered Ramsey numbers. For so-called
monotone cycles we compute their ordered Ramsey numbers exactly. This result
implies exact formulas for geometric Ramsey numbers of cycles introduced by
K\'arolyi, Pach, T\'oth, and Valtr.Comment: 29 pages, 13 figures, to appear in Electronic Journal of
Combinatoric
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
On deficiency problems for graphs
Motivated by analogous questions in the setting of Steiner triple systems and
Latin squares, Nenadov, Sudakov and Wagner [Completion and deficiency problems,
Journal of Combinatorial Theory Series B, 2020] recently introduced the notion
of graph deficiency. Given a global spanning property and a graph
, the deficiency of the graph with respect to the
property is the smallest non-negative integer such that the
join has property . In particular, Nenadov, Sudakov and
Wagner raised the question of determining how many edges an -vertex graph
needs to ensure contains a -factor (for any fixed ).
In this paper we resolve their problem fully. We also give an analogous result
which forces to contain any fixed bipartite -vertex graph of
bounded degree and small bandwidth.Comment: 11 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
Tilings in randomly perturbed dense graphs
A perfect -tiling in a graph is a collection of vertex-disjoint copies
of a graph in that together cover all the vertices in . In this
paper we investigate perfect -tilings in a random graph model introduced by
Bohman, Frieze and Martin in which one starts with a dense graph and then adds
random edges to it. Specifically, for any fixed graph , we determine the
number of random edges required to add to an arbitrary graph of linear minimum
degree in order to ensure the resulting graph contains a perfect -tiling
with high probability. Our proof utilises Szemer\'edi's Regularity lemma as
well as a special case of a result of Koml\'os concerning almost perfect
-tilings in dense graphs.Comment: 19 pages, to appear in CP
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