413 research outputs found
Hamilton decompositions of regular expanders: a proof of Kelly's conjecture for large tournaments
A long-standing conjecture of Kelly states that every regular tournament on n
vertices can be decomposed into (n-1)/2 edge-disjoint Hamilton cycles. We prove
this conjecture for large n. In fact, we prove a far more general result, based
on our recent concept of robust expansion and a new method for decomposing
graphs. We show that every sufficiently large regular digraph G on n vertices
whose degree is linear in n and which is a robust outexpander has a
decomposition into edge-disjoint Hamilton cycles. This enables us to obtain
numerous further results, e.g. as a special case we confirm a conjecture of
Erdos on packing Hamilton cycles in random tournaments. As corollaries to the
main result, we also obtain several results on packing Hamilton cycles in
undirected graphs, giving e.g. the best known result on a conjecture of
Nash-Williams. We also apply our result to solve a problem on the domination
ratio of the Asymmetric Travelling Salesman problem, which was raised e.g. by
Glover and Punnen as well as Alon, Gutin and Krivelevich.Comment: new version includes a standalone version of the `robust
decomposition lemma' for application in subsequent paper
Hamilton decompositions of regular tournaments
We show that every sufficiently large regular tournament can almost
completely be decomposed into edge-disjoint Hamilton cycles. More precisely,
for each \eta>0 every regular tournament G of sufficiently large order n
contains at least (1/2-\eta)n edge-disjoint Hamilton cycles. This gives an
approximate solution to a conjecture of Kelly from 1968. Our result also
extends to almost regular tournaments.Comment: 38 pages, 2 figures. Added section sketching how we can extend our
main result. To appear in the Proceedings of the LM
Proof of the 1-factorization and Hamilton decomposition conjectures III: approximate decompositions
In a sequence of four papers, we prove the following results (via a unified
approach) for all sufficiently large :
(i) [1-factorization conjecture] Suppose that is even and . Then every -regular graph on vertices has a
decomposition into perfect matchings. Equivalently, .
(ii) [Hamilton decomposition conjecture] Suppose that . Then every -regular graph on vertices has a decomposition
into Hamilton cycles and at most one perfect matching.
(iii) We prove an optimal result on the number of edge-disjoint Hamilton
cycles in a graph of given minimum degree.
According to Dirac, (i) was first raised in the 1950s. (ii) and (iii) answer
questions of Nash-Williams from 1970. The above bounds are best possible. In
the current paper, we show the following: suppose that is close to a
complete balanced bipartite graph or to the union of two cliques of equal size.
If we are given a suitable set of path systems which cover a set of
`exceptional' vertices and edges of , then we can extend these path systems
into an approximate decomposition of into Hamilton cycles (or perfect
matchings if appropriate).Comment: We originally split the proof into four papers, of which this was the
third paper. We have now combined this series into a single publication
[arXiv:1401.4159v2], which will appear in the Memoirs of the AMS. 29 pages, 2
figure
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 decompositions of regular expanders: applications
In a recent paper, we showed that every sufficiently large regular digraph G
on n vertices whose degree is linear in n and which is a robust outexpander has
a decomposition into edge-disjoint Hamilton cycles. The main consequence of
this theorem is that every regular tournament on n vertices can be decomposed
into (n-1)/2 edge-disjoint Hamilton cycles, whenever n is sufficiently large.
This verified a conjecture of Kelly from 1968. In this paper, we derive a
number of further consequences of our result on robust outexpanders, the main
ones are the following: (i) an undirected analogue of our result on robust
outexpanders; (ii) best possible bounds on the size of an optimal packing of
edge-disjoint Hamilton cycles in a graph of minimum degree d for a large range
of values for d. (iii) a similar result for digraphs of given minimum
semidegree; (iv) an approximate version of a conjecture of Nash-Williams on
Hamilton decompositions of dense regular graphs; (v) the observation that dense
quasi-random graphs are robust outexpanders; (vi) a verification of the `very
dense' case of a conjecture of Frieze and Krivelevich on packing edge-disjoint
Hamilton cycles in random graphs; (vii) a proof of a conjecture of Erdos on the
size of an optimal packing of edge-disjoint Hamilton cycles in a random
tournament.Comment: final version, to appear in J. Combinatorial Theory
Approximate Hamilton decompositions of robustly expanding regular digraphs
We show that every sufficiently large r-regular digraph G which has linear
degree and is a robust outexpander has an approximate decomposition into
edge-disjoint Hamilton cycles, i.e. G contains a set of r-o(r) edge-disjoint
Hamilton cycles. Here G is a robust outexpander if for every set S which is not
too small and not too large, the `robust' outneighbourhood of S is a little
larger than S. This generalises a result of K\"uhn, Osthus and Treglown on
approximate Hamilton decompositions of dense regular oriented graphs. It also
generalises a result of Frieze and Krivelevich on approximate Hamilton
decompositions of quasirandom (di)graphs. In turn, our result is used as a tool
by K\"uhn and Osthus to prove that any sufficiently large r-regular digraph G
which has linear degree and is a robust outexpander even has a Hamilton
decomposition.Comment: Final version, published in SIAM Journal Discrete Mathematics. 44
pages, 2 figure
Counting Hamilton decompositions of oriented graphs
A Hamilton cycle in a directed graph G is a cycle that passes through every vertex of G. A Hamilton decomposition of G is a partition of its edge set into disjoint Hamilton cycles. In the late 60s, Kelly conjectured that every regular tournament has a Hamilton decomposition. This conjecture was recently settled for large tournaments by Kühn and Osthus [13], who proved more generally that every r-regular n-vertex oriented graph G (without antiparallel edges) with r=cn for some fixed c>3/8 has a Hamilton decomposition, provided n=n(c) is sufficiently large. In this article, we address the natural question of estimating the number of such decompositions of G and show that this number is n(1−o(1))cn^2. In addition, we also obtain a new and much simpler proof for the approximate version of Kelly’s conjecture
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