823 research outputs found
Hamiltonian degree sequences in digraphs
We show that for each \eta>0 every digraph G of sufficiently large order n is
Hamiltonian if its out- and indegree sequences d^+_1\le ... \le d^+_n and d^-
_1 \le ... \le d^-_n satisfy
(i) d^+_i \geq i+ \eta n or d^-_{n-i- \eta n} \geq n-i and
(ii) d^-_i \geq i+ \eta n or d^+_{n-i- \eta n} \geq n-i for all i < n/2.
This gives an approximate solution to a problem of Nash-Williams concerning a
digraph analogue of Chv\'atal's theorem. In fact, we prove the stronger result
that such digraphs G are pancyclic.Comment: 17 pages, 2 figures. Section added which includes a proof of a
conjecture of Thomassen for large tournaments. To appear in JCT
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
A semi-exact degree condition for Hamilton cycles in digraphs
The paper is concerned with directed versions of Posa's theorem and Chvatal's
theorem on Hamilton cycles in graphs.
We show that for each a>0, every digraph G of sufficiently large order n
whose outdegree and indegree sequences d_1^+ \leq ... \leq d_n^+ and d_1^- \leq
>... \leq d_n^- satisfy d_i^+, d_i^- \geq min{i + a n, n/2} is Hamiltonian. In
fact, we can weaken these assumptions to
(i) d_i^+ \geq min{i + a n, n/2} or d^-_{n - i - a n} \geq n-i; (ii) d_i^-
\geq min{i + a n, n/2} or d^+_{n - i - a n} \geq n-i; and still deduce that G
is Hamiltonian. This provides an approximate version of a conjecture of
Nash-Williams from 1975 and improves a previous result of K\"uhn, Osthus and
Treglown
Embedding large subgraphs into dense graphs
What conditions ensure that a graph G contains some given spanning subgraph
H? The most famous examples of results of this kind are probably Dirac's
theorem on Hamilton cycles and Tutte's theorem on perfect matchings. Perfect
matchings are generalized by perfect F-packings, where instead of covering all
the vertices of G by disjoint edges, we want to cover G by disjoint copies of a
(small) graph F. It is unlikely that there is a characterization of all graphs
G which contain a perfect F-packing, so as in the case of Dirac's theorem it
makes sense to study conditions on the minimum degree of G which guarantee a
perfect F-packing.
The Regularity lemma of Szemeredi and the Blow-up lemma of Komlos, Sarkozy
and Szemeredi have proved to be powerful tools in attacking such problems and
quite recently, several long-standing problems and conjectures in the area have
been solved using these. In this survey, we give an outline of recent progress
(with our main emphasis on F-packings, Hamiltonicity problems and tree
embeddings) and describe some of the methods involved
Counting Hamilton cycles in sparse random directed graphs
Let D(n,p) be the random directed graph on n vertices where each of the
n(n-1) possible arcs is present independently with probability p. A celebrated
result of Frieze shows that if then D(n,p) typically
has a directed Hamilton cycle, and this is best possible. In this paper, we
obtain a strengthening of this result, showing that under the same condition,
the number of directed Hamilton cycles in D(n,p) is typically
. We also prove a hitting-time version of this statement,
showing that in the random directed graph process, as soon as every vertex has
in-/out-degrees at least 1, there are typically
directed Hamilton cycles
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