1,260 research outputs found
Tilings in vertex ordered graphs
Over recent years there has been much interest in both Tur\'an and Ramsey
properties of vertex ordered graphs. In this paper we initiate the study of
embedding spanning structures into vertex ordered graphs. In particular, we
introduce a general framework for approaching the problem of determining the
minimum degree threshold for forcing a perfect -tiling in an ordered graph.
In the (unordered) graph setting, this problem was resolved by K\"uhn and
Osthus [The minimum degree threshold for perfect graph packings, Combinatorica,
2009]. We use our general framework to resolve the perfect -tiling problem
for all ordered graphs of interval chromatic number . Already in this
restricted setting the class of extremal examples is richer than in the
unordered graph problem. In the process of proving our results, novel
approaches to both the regularity and absorbing methods are developed.Comment: 22 pages, updated to address referee comment
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
Perfect packings with complete graphs minus an edge
Let K_r^- denote the graph obtained from K_r by deleting one edge. We show
that for every integer r\ge 4 there exists an integer n_0=n_0(r) such that
every graph G whose order n\ge n_0 is divisible by r and whose minimum degree
is at least (1-1/chi_{cr}(K_r^-))n contains a perfect K_r^- packing, i.e. a
collection of disjoint copies of K_r^- which covers all vertices of G. Here
chi_{cr}(K_r^-)=r(r-2)/(r-1) is the critical chromatic number of K_r^-. The
bound on the minimum degree is best possible and confirms a conjecture of
Kawarabayashi for large n
On perfect packings in dense graphs
We say that a graph G has a perfect H-packing if there exists a set of
vertex-disjoint copies of H which cover all the vertices in G. We consider
various problems concerning perfect H-packings: Given positive integers n, r,
D, we characterise the edge density threshold that ensures a perfect
K_r-packing in any graph G on n vertices and with minimum degree at least D. We
also give two conjectures concerning degree sequence conditions which force a
graph to contain a perfect H-packing. Other related embedding problems are also
considered. Indeed, we give a structural result concerning K_r-free graphs that
satisfy a certain degree sequence condition.Comment: 18 pages, 1 figure. Electronic Journal of Combinatorics 20(1) (2013)
#P57. This version contains an open problem sectio
Perfect Packings in Quasirandom Hypergraphs II
For each of the notions of hypergraph quasirandomness that have been studied,
we identify a large class of hypergraphs F so that every quasirandom hypergraph
H admits a perfect F-packing. An informal statement of a special case of our
general result for 3-uniform hypergraphs is as follows. Fix an integer r >= 4
and 0<p<1. Suppose that H is an n-vertex triple system with r|n and the
following two properties:
* for every graph G with V(G)=V(H), at least p proportion of the triangles in
G are also edges of H,
* for every vertex x of H, the link graph of x is a quasirandom graph with
density at least p.
Then H has a perfect -packing. Moreover, we show that neither
hypotheses above can be weakened, so in this sense our result is tight. A
similar conclusion for this special case can be proved by Keevash's hypergraph
blowup lemma, with a slightly stronger hypothesis on H.Comment: 17 page
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
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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