342 research outputs found
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 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: applications
In a recent paper, we showed 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. The main consequence of
this theorem is that every regular tournament on n vertices can be decomposed
into (n-1)/2 edge-disjoint Hamilton cycles, whenever n is sufficiently large.
This verified a conjecture of Kelly from 1968. In this paper, we derive a
number of further consequences of our result on robust outexpanders, the main
ones are the following: (i) an undirected analogue of our result on robust
outexpanders; (ii) best possible bounds on the size of an optimal packing of
edge-disjoint Hamilton cycles in a graph of minimum degree d for a large range
of values for d. (iii) a similar result for digraphs of given minimum
semidegree; (iv) an approximate version of a conjecture of Nash-Williams on
Hamilton decompositions of dense regular graphs; (v) the observation that dense
quasi-random graphs are robust outexpanders; (vi) a verification of the `very
dense' case of a conjecture of Frieze and Krivelevich on packing edge-disjoint
Hamilton cycles in random graphs; (vii) a proof of a conjecture of Erdos on the
size of an optimal packing of edge-disjoint Hamilton cycles in a random
tournament.Comment: final version, to appear in J. Combinatorial Theory
Optimal Hamilton covers and linear arboricity for random graphs
In his seminal 1976 paper, P\'osa showed that for all , the
binomial random graph is with high probability Hamiltonian. This leads
to the following natural questions, which have been extensively studied: How
well is it typically possible to cover all edges of with Hamilton
cycles? How many cycles are necessary? In this paper we show that for , we can cover with precisely
Hamilton cycles. Our result is clearly best possible
both in terms of the number of required cycles, and the asymptotics of the edge
probability , since it starts working at the weak threshold needed for
Hamiltonicity. This resolves a problem of Glebov, Krivelevich and Szab\'o, and
improves upon previous work of Hefetz, K\"uhn, Lapinskas and Osthus, and of
Ferber, Kronenberg and Long, essentially closing a long line of research on
Hamiltonian packing and covering problems in random graphs.Comment: 13 page
A bandwidth theorem for approximate decompositions
We provide a degree condition on a regular -vertex graph which ensures
the existence of a near optimal packing of any family of bounded
degree -vertex -chromatic separable graphs into . In general, this
degree condition is best possible.
Here a graph is separable if it has a sublinear separator whose removal
results in a set of components of sublinear size. Equivalently, the
separability condition can be replaced by that of having small bandwidth. Thus
our result can be viewed as a version of the bandwidth theorem of B\"ottcher,
Schacht and Taraz in the setting of approximate decompositions.
More precisely, let be the infimum over all
ensuring an approximate -decomposition of any sufficiently large regular
-vertex graph of degree at least . Now suppose that is an
-vertex graph which is close to -regular for some and suppose that is a sequence of bounded
degree -vertex -chromatic separable graphs with . We show that there is an edge-disjoint packing of
into .
If the are bipartite, then is sufficient. In
particular, this yields an approximate version of the tree packing conjecture
in the setting of regular host graphs of high degree. Similarly, our result
implies approximate versions of the Oberwolfach problem, the Alspach problem
and the existence of resolvable designs in the setting of regular host graphs
of high degree.Comment: Final version, to appear in the Proceedings of the London
Mathematical Societ
Optimal path and cycle decompositions of dense quasirandom graphs
Motivated by longstanding conjectures regarding decompositions of graphs into
paths and cycles, we prove the following optimal decomposition results for
random graphs. Let be constant and let . Let be
the number of odd degree vertices in . Then a.a.s. the following hold:
(i) can be decomposed into cycles and a
matching of size .
(ii) can be decomposed into
paths.
(iii) can be decomposed into linear forests.
Each of these bounds is best possible. We actually derive (i)--(iii) from
`quasirandom' versions of our results. In that context, we also determine the
edge chromatic number of a given dense quasirandom graph of even order. For all
these results, our main tool is a result on Hamilton decompositions of robust
expanders by K\"uhn and Osthus.Comment: Some typos from the first version have been correcte
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