2,113 research outputs found
Tight cycles in hypergraphs
We apply a recent version of the Strong Hypergraph Regularity Lemma(see [1], [2]) to prove two new results on tight cycles in k-uniform hypergraphs. The first result is an extension of the Erdos-Gallai Theorem for graphs: For every > 0, every sufficiently large k-uniform hypergraph on n vertices with at least edges contains a tight cycle of length @n for any @ 2 [0; 1]. Our second result concerns k-partite k-uniform hypergraphs with partition classes of size n and for each @ 2 (0; 1) provides an asymptotically optimal minimum codegree requirement for the hypergraph to contain a cycle of length @kn
Tight Hamilton Cycles in Random Uniform Hypergraphs
In this paper we show that is the sharp threshold for the existence of
tight Hamilton cycles in random -uniform hypergraphs, for all . When
we show that is an asymptotic threshold. We also determine
thresholds for the existence of other types of Hamilton cycles.Comment: 9 pages. Updated to add materia
Finding tight Hamilton cycles in random hypergraphs faster
In an -uniform hypergraph on vertices a tight Hamilton cycle consists
of edges such that there exists a cyclic ordering of the vertices where the
edges correspond to consecutive segments of vertices. We provide a first
deterministic polynomial time algorithm, which finds a.a.s. tight Hamilton
cycles in random -uniform hypergraphs with edge probability at least . Our result partially answers a question of Dudek and Frieze [Random
Structures & Algorithms 42 (2013), 374-385] who proved that tight Hamilton
cycles exists already for for and for
using a second moment argument. Moreover our algorithm is superior to
previous results of Allen, B\"ottcher, Kohayakawa and Person [Random Structures
& Algorithms 46 (2015), 446-465] and Nenadov and \v{S}kori\'c
[arXiv:1601.04034] in various ways: the algorithm of Allen et al. is a
randomised polynomial time algorithm working for edge probabilities , while the algorithm of Nenadov and \v{S}kori\'c is a
randomised quasipolynomial time algorithm working for edge probabilities .Comment: 17 page
Packing tight Hamilton cycles in 3-uniform hypergraphs
Let H be a 3-uniform hypergraph with N vertices. A tight Hamilton cycle C
\subset H is a collection of N edges for which there is an ordering of the
vertices v_1, ..., v_N such that every triple of consecutive vertices {v_i,
v_{i+1}, v_{i+2}} is an edge of C (indices are considered modulo N). We develop
new techniques which enable us to prove that under certain natural
pseudo-random conditions, almost all edges of H can be covered by edge-disjoint
tight Hamilton cycles, for N divisible by 4. Consequently, we derive the
corollary that random 3-uniform hypergraphs can be almost completely packed
with tight Hamilton cycles w.h.p., for N divisible by 4 and P not too small.
Along the way, we develop a similar result for packing Hamilton cycles in
pseudo-random digraphs with even numbers of vertices.Comment: 31 pages, 1 figur
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
Cycle decompositions in k-uniform hypergraphs
We show that k-uniform hypergraphs on n vertices whose codegree is at least (2/3+o(1))n can be decomposed into tight cycles, subject to the trivial divisibility conditions. As a corollary, we show those graphs contain tight Euler tours as well. In passing, we also investigate decompositions into tight paths.In addition, we also prove an alternative condition for building absorbers for edge-decompositions of arbitrary k-uniform hypergraphs, which should be of independent interest
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