168 research outputs found
An Ore-type theorem for perfect packings in graphs
We say that a graph G has a perfect H-packing (also called an H-factor) if
there exists a set of disjoint copies of H in G which together cover all the
vertices of G. Given a graph H, we determine, asymptotically, the Ore-type
degree condition which ensures that a graph G has a perfect H-packing. More
precisely, let \delta_{\rm Ore} (H,n) be the smallest number k such that every
graph G whose order n is divisible by |H| and with d(x)+d(y)\geq k for all
non-adjacent x \not = y \in V(G) contains a perfect H-packing. We determine
\lim_{n\to \infty} \delta_{\rm Ore} (H,n)/n.Comment: 23 pages, 1 figure. Extra examples and a sketch proof of Theorem 4
added. To appear in the SIAM Journal on Discrete Mathematic
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
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
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 spanning bipartite graphs of small bandwidth
Boettcher, Schacht and Taraz gave a condition on the minimum degree of a
graph G on n vertices that ensures G contains every r-chromatic graph H on n
vertices of bounded degree and of bandwidth o(n), thereby proving a conjecture
of Bollobas and Komlos. We strengthen this result in the case when H is
bipartite. Indeed, we give an essentially best-possible condition on the degree
sequence of a graph G on n vertices that forces G to contain every bipartite
graph H on n vertices of bounded degree and of bandwidth o(n). This also
implies an Ore-type result. In fact, we prove a much stronger result where the
condition on G is relaxed to a certain robust expansion property. Our result
also confirms the bipartite case of a conjecture of Balogh, Kostochka and
Treglown concerning the degree sequence of a graph which forces a perfect
H-packing.Comment: 23 pages, file updated, to appear in Combinatorics, Probability and
Computin
- …