68 research outputs found
The minimum vertex degree for an almost-spanning tight cycle in a -uniform hypergraph
We prove that any -uniform hypergraph whose minimum vertex degree is at
least admits an almost-spanning
tight cycle, that is, a tight cycle leaving vertices uncovered. The
bound on the vertex degree is asymptotically best possible. Our proof uses the
hypergraph regularity method, and in particular a recent version of the
hypergraph regularity lemma proved by Allen, B\"ottcher, Cooley and Mycroft.Comment: 10 pages. arXiv admin note: text overlap with arXiv:1411.495
Hamilton cycles in hypergraphs below the Dirac threshold
We establish a precise characterisation of -uniform hypergraphs with
minimum codegree close to which contain a Hamilton -cycle. As an
immediate corollary we identify the exact Dirac threshold for Hamilton
-cycles in -uniform hypergraphs. Moreover, by derandomising the proof of
our characterisation we provide a polynomial-time algorithm which, given a
-uniform hypergraph with minimum codegree close to , either finds a
Hamilton -cycle in or provides a certificate that no such cycle exists.
This surprising result stands in contrast to the graph setting, in which below
the Dirac threshold it is NP-hard to determine if a graph is Hamiltonian. We
also consider tight Hamilton cycles in -uniform hypergraphs for , giving a series of reductions to show that it is NP-hard to determine
whether a -uniform hypergraph with minimum degree contains a tight Hamilton cycle. It is therefore
unlikely that a similar characterisation can be obtained for tight Hamilton
cycles.Comment: v2: minor revisions in response to reviewer comments, most pseudocode
and details of the polynomial time reduction moved to the appendix which will
not appear in the printed version of the paper. To appear in Journal of
Combinatorial Theory, Series
Hamilton cycles in quasirandom hypergraphs
We show that, for a natural notion of quasirandomness in -uniform
hypergraphs, any quasirandom -uniform hypergraph on vertices with
constant edge density and minimum vertex degree contains a
loose Hamilton cycle. We also give a construction to show that a -uniform
hypergraph satisfying these conditions need not contain a Hamilton -cycle
if divides . The remaining values of form an interesting
open question.Comment: 18 pages. Accepted for publication in Random Structures & Algorithm
On powers of tight Hamilton cycles in randomly perturbed hypergraphs
We show that for , and , there exists
such that if and is a -uniform hypergraph
on vertices with minimum codegree at least , then asymptotically
almost surely the union contains the power of a
tight Hamilton cycle. The bound on is optimal up to the value of
and this answers a question of Bedenknecht, Han, Kohayakawa and
Mota
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
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