8,392 research outputs found
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
Pseudo-random graphs
Random graphs have proven to be one of the most important and fruitful
concepts in modern Combinatorics and Theoretical Computer Science. Besides
being a fascinating study subject for their own sake, they serve as essential
instruments in proving an enormous number of combinatorial statements, making
their role quite hard to overestimate. Their tremendous success serves as a
natural motivation for the following very general and deep informal questions:
what are the essential properties of random graphs? How can one tell when a
given graph behaves like a random graph? How to create deterministically graphs
that look random-like? This leads us to a concept of pseudo-random graphs and
the aim of this survey is to provide a systematic treatment of this concept.Comment: 50 page
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
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
The number of Hamiltonian decompositions of regular graphs
A Hamilton cycle in a graph is a cycle passing through every vertex
of . A Hamiltonian decomposition of is a partition of its edge
set into disjoint Hamilton cycles. One of the oldest results in graph theory is
Walecki's theorem from the 19th century, showing that a complete graph on
an odd number of vertices has a Hamiltonian decomposition. This result was
recently greatly extended by K\"{u}hn and Osthus. They proved that every
-regular -vertex graph with even degree for some fixed
has a Hamiltonian decomposition, provided is sufficiently
large. In this paper we address the natural question of estimating ,
the number of such decompositions of . Our main result is that
. In particular, the number of Hamiltonian
decompositions of is
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
Hamilton cycles in highly connected and expanding graphs
In this paper we prove a sufficient condition for the existence of a Hamilton
cycle, which is applicable to a wide variety of graphs, including relatively
sparse graphs. In contrast to previous criteria, ours is based on only two
properties: one requiring expansion of ``small'' sets, the other ensuring the
existence of an edge between any two disjoint ``large'' sets. We also discuss
applications in positional games, random graphs and extremal graph theory.Comment: 19 page
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