12,176 research outputs found
Optimal covers with Hamilton cycles in random graphs
A packing of a graph G with Hamilton cycles is a set of edge-disjoint
Hamilton cycles in G. Such packings have been studied intensively and recent
results imply that a largest packing of Hamilton cycles in G_n,p a.a.s. has
size \lfloor delta(G_n,p) /2 \rfloor. Glebov, Krivelevich and Szab\'o recently
initiated research on the `dual' problem, where one asks for a set of Hamilton
cycles covering all edges of G. Our main result states that for log^{117}n / n
< p < 1-n^{-1/8}, a.a.s. the edges of G_n,p can be covered by \lceil
Delta(G_n,p)/2 \rceil Hamilton cycles. This is clearly optimal and improves an
approximate result of Glebov, Krivelevich and Szab\'o, which holds for p >
n^{-1+\eps}. Our proof is based on a result of Knox, K\"uhn and Osthus on
packing Hamilton cycles in pseudorandom graphs.Comment: final version of paper (to appear in Combinatorica
Limited packings of closed neighbourhoods in graphs
The k-limited packing number, , of a graph , introduced by
Gallant, Gunther, Hartnell, and Rall, is the maximum cardinality of a set
of vertices of such that every vertex of has at most elements of
in its closed neighbourhood. The main aim in this paper is to prove the
best-possible result that if is a cubic graph, then , improving the previous lower bound given by Gallant, \emph{et al.}
In addition, we construct an infinite family of graphs to show that lower
bounds given by Gagarin and Zverovich are asymptotically best-possible, up to a
constant factor, when is fixed and tends to infinity. For
tending to infinity and tending to infinity sufficiently
quickly, we give an asymptotically best-possible lower bound for ,
improving previous bounds
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
The existence of designs via iterative absorption: hypergraph -designs for arbitrary
We solve the existence problem for -designs for arbitrary -uniform
hypergraphs~. This implies that given any -uniform hypergraph~, the
trivially necessary divisibility conditions are sufficient to guarantee a
decomposition of any sufficiently large complete -uniform hypergraph into
edge-disjoint copies of~, which answers a question asked e.g.~by Keevash.
The graph case was proved by Wilson in 1975 and forms one of the
cornerstones of design theory. The case when~ is complete corresponds to the
existence of block designs, a problem going back to the 19th century, which was
recently settled by Keevash. In particular, our argument provides a new proof
of the existence of block designs, based on iterative absorption (which employs
purely probabilistic and combinatorial methods).
Our main result concerns decompositions of hypergraphs whose clique
distribution fulfills certain regularity constraints. Our argument allows us to
employ a `regularity boosting' process which frequently enables us to satisfy
these constraints even if the clique distribution of the original hypergraph
does not satisfy them. This enables us to go significantly beyond the setting
of quasirandom hypergraphs considered by Keevash. In particular, we obtain a
resilience version and a decomposition result for hypergraphs of large minimum
degree.Comment: This version combines the two manuscripts `The existence of designs
via iterative absorption' (arXiv:1611.06827v1) and the subsequent `Hypergraph
F-designs for arbitrary F' (arXiv:1706.01800) into a single paper, which will
appear in the Memoirs of the AM
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
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