9,591 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
Seiberg-Witten-Floer homology of a surface times a circle for non-torsion spin-c structures
We determine the Seiberg-Witten-Floer homology groups of the three-manifold
which is the product of a surface of genus times the circle,
together with its ring structure, for spin-c structures which are non-trivial
on the three-manifold. We give applications to computing Seiberg-Witten
invariants of four-manifolds which are connected sums along surfaces and also
we reprove the higher type adjunction inequalities previously obtained by
Oszv\'ath and Szab\'o.Comment: 26 pages, no figures, Latex2e, to appear in Math. Nac
Tracially sequentially-split -homomorphisms between -algebras
We define a tracial analogue of the sequentially split -homomorphism
between -algebras of Barlak and Szab\'{o} and show that several important
approximation properties related to the classification theory of -algebras
pass from the target algebra to the domain algebra. Then we show that the
tracial Rokhlin property of the finite group action on a -algebra
gives rise to a tracial version of sequentially split -homomorphism from
to and the tracial Rokhlin property of an
inclusion -algebras with a conditional expectation of a finite Watatani index generates a tracial version of sequentially split
map. By doing so, we provide a unified approach to permanence properties
related to tracial Rokhlin property of operator algebras.Comment: A serious flaw in Definition 2.6 has been notified to the authors. We
fix our definition and accordingly change statements in subsequent
propositions and theorems. Moreover, a gap in the proof of Theorem 2.25 is
fixed. We note our appreciation for such helpful comments in Acknowledgements
section. Some typos are also caught. We hope that it is fina
Complete subgraphs in a multipartite graph
In 1975 Bollob\'as, Erd\H os, and Szemer\'edi asked the following question:
given positive integers with , what is the largest
minimum degree among all -partite graphs with parts of size
and which do not contain a copy of ? The case has
attracted a lot of attention and was fully resolved by Haxell and Szab\'{o},
and Szab\'{o} and Tardos in 2006. In this paper we investigate the case
of the problem, which has remained dormant for over forty years. We resolve the
problem exactly in the case when , and up to an additive
constant for many other cases, including when . Our
approach utilizes a connection to the related problem of determining the
maximum of the minimum degrees among the family of balanced -partite
-vertex graphs of chromatic number at most
Powers of Hamilton cycles in pseudorandom graphs
We study the appearance of powers of Hamilton cycles in pseudorandom graphs,
using the following comparatively weak pseudorandomness notion. A graph is
-pseudorandom if for all disjoint and with and we have
. We prove that for all there is an
such that an -pseudorandom graph on
vertices with minimum degree at least contains the square of a
Hamilton cycle. In particular, this implies that -graphs with
contain the square of a Hamilton cycle, and thus
a triangle factor if is a multiple of . This improves on a result of
Krivelevich, Sudakov and Szab\'o [Triangle factors in sparse pseudo-random
graphs, Combinatorica 24 (2004), no. 3, 403--426].
We also extend our result to higher powers of Hamilton cycles and establish
corresponding counting versions.Comment: 30 pages, 1 figur
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