542 research outputs found
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
Proof of the 1-factorization and Hamilton decomposition conjectures IV: exceptional systems for the two cliques case
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 prove results on the decomposition of sparse graphs into
path systems. These are used in the proof of (i) and (ii) in the case when
is close to the union of two disjoint cliques.Comment: We originally split the proof into four papers, of which this was the
fourth paper. We have now combined this series into a single publication
[arXiv:1401.4159v2], which will appear in the Memoirs of the AMS. 37 page
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
Optimal path and cycle decompositions of dense quasirandom graphs
Motivated by longstanding conjectures regarding decompositions of graphs into
paths and cycles, we prove the following optimal decomposition results for
random graphs. Let be constant and let . Let be
the number of odd degree vertices in . Then a.a.s. the following hold:
(i) can be decomposed into cycles and a
matching of size .
(ii) can be decomposed into
paths.
(iii) can be decomposed into linear forests.
Each of these bounds is best possible. We actually derive (i)--(iii) from
`quasirandom' versions of our results. In that context, we also determine the
edge chromatic number of a given dense quasirandom graph of even order. For all
these results, our main tool is a result on Hamilton decompositions of robust
expanders by K\"uhn and Osthus.Comment: Some typos from the first version have been correcte
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
Pseudo-modularity and Iwasawa theory
We prove, assuming Greenberg's conjecture, that the ordinary eigencurve is
Gorenstein at an intersection point between the Eisenstein family and the
cuspidal locus. As a corollary, we obtain new results on Sharifi's conjecture.
This result is achieved by constructing a universal ordinary pseudodeformation
ring and proving an result.Comment: Changes to section 5.9; typos corrected. To appear in Amer. J. Math.
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