241 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
Hamilton decompositions of regular expanders: a proof of Kelly's conjecture for large tournaments
A long-standing conjecture of Kelly states that every regular tournament on n
vertices can be decomposed into (n-1)/2 edge-disjoint Hamilton cycles. We prove
this conjecture for large n. In fact, we prove a far more general result, based
on our recent concept of robust expansion and a new method for decomposing
graphs. We show that every sufficiently large regular digraph G on n vertices
whose degree is linear in n and which is a robust outexpander has a
decomposition into edge-disjoint Hamilton cycles. This enables us to obtain
numerous further results, e.g. as a special case we confirm a conjecture of
Erdos on packing Hamilton cycles in random tournaments. As corollaries to the
main result, we also obtain several results on packing Hamilton cycles in
undirected graphs, giving e.g. the best known result on a conjecture of
Nash-Williams. We also apply our result to solve a problem on the domination
ratio of the Asymmetric Travelling Salesman problem, which was raised e.g. by
Glover and Punnen as well as Alon, Gutin and Krivelevich.Comment: new version includes a standalone version of the `robust
decomposition lemma' for application in subsequent paper
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
Approximate Hamilton decompositions of robustly expanding regular digraphs
We show that every sufficiently large r-regular digraph G which has linear
degree and is a robust outexpander has an approximate decomposition into
edge-disjoint Hamilton cycles, i.e. G contains a set of r-o(r) edge-disjoint
Hamilton cycles. Here G is a robust outexpander if for every set S which is not
too small and not too large, the `robust' outneighbourhood of S is a little
larger than S. This generalises a result of K\"uhn, Osthus and Treglown on
approximate Hamilton decompositions of dense regular oriented graphs. It also
generalises a result of Frieze and Krivelevich on approximate Hamilton
decompositions of quasirandom (di)graphs. In turn, our result is used as a tool
by K\"uhn and Osthus to prove that any sufficiently large r-regular digraph G
which has linear degree and is a robust outexpander even has a Hamilton
decomposition.Comment: Final version, published in SIAM Journal Discrete Mathematics. 44
pages, 2 figure
Hamilton decompositions of regular tournaments
We show that every sufficiently large regular tournament can almost
completely be decomposed into edge-disjoint Hamilton cycles. More precisely,
for each \eta>0 every regular tournament G of sufficiently large order n
contains at least (1/2-\eta)n edge-disjoint Hamilton cycles. This gives an
approximate solution to a conjecture of Kelly from 1968. Our result also
extends to almost regular tournaments.Comment: 38 pages, 2 figures. Added section sketching how we can extend our
main result. To appear in the Proceedings of the LM
Edge-disjoint Hamilton cycles in graphs
In this paper we give an approximate answer to a question of Nash-Williams
from 1970: we show that for every \alpha > 0, every sufficiently large graph on
n vertices with minimum degree at least (1/2 + \alpha)n contains at least n/8
edge-disjoint Hamilton cycles. More generally, we give an asymptotically best
possible answer for the number of edge-disjoint Hamilton cycles that a graph G
with minimum degree \delta must have. We also prove an approximate version of
another long-standing conjecture of Nash-Williams: we show that for every
\alpha > 0, every (almost) regular and sufficiently large graph on n vertices
with minimum degree at least can be almost decomposed into
edge-disjoint Hamilton cycles.Comment: Minor Revisio
Proof of a tournament partition conjecture and an application to 1-factors with prescribed cycle lengths
In 1982 Thomassen asked whether there exists an integer f(k,t) such that
every strongly f(k,t)-connected tournament T admits a partition of its vertex
set into t vertex classes V_1,...,V_t such that for all i the subtournament
T[V_i] induced on T by V_i is strongly k-connected. Our main result implies an
affirmative answer to this question. In particular we show that f(k,t) = O(k^7
t^4) suffices. As another application of our main result we give an affirmative
answer to a question of Song as to whether, for any integer t, there exists an
integer h(t) such that every strongly h(t)-connected tournament has a 1-factor
consisting of t vertex-disjoint cycles of prescribed lengths. We show that h(t)
= O(t^5) suffices.Comment: final version, to appear in Combinatoric
A Dirac type result on Hamilton cycles in oriented graphs
We show that for each \alpha>0 every sufficiently large oriented graph G with
\delta^+(G),\delta^-(G)\ge 3|G|/8+ \alpha |G| contains a Hamilton cycle. This
gives an approximate solution to a problem of Thomassen. In fact, we prove the
stronger result that G is still Hamiltonian if
\delta(G)+\delta^+(G)+\delta^-(G)\geq 3|G|/2 + \alpha |G|. Up to the term
\alpha |G| this confirms a conjecture of H\"aggkvist. We also prove an Ore-type
theorem for oriented graphs.Comment: Added an Ore-type resul
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