41 research outputs found
Pancyclicity of highly connected graphs
A well-known result due to Chvat\'al and Erd\H{o}s (1972) asserts that, if a
graph satisfies , where is the
vertex-connectivity of , then has a Hamilton cycle. We prove a similar
result implying that a graph is pancyclic, namely it contains cycles of all
lengths between and : if is large and ,
then is pancyclic. This confirms a conjecture of Jackson and Ordaz (1990)
for large graphs, and improves upon a very recent result of Dragani\'c,
Munh\'a-Correia, and Sudakov.Comment: 31 pages, 11 figure
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
Fast Strategies in Waiter-Client Games on
Waiter-Client games are played on some hypergraph , where
denotes the family of winning sets. For some bias , during
each round of such a game Waiter offers to Client elements of , of
which Client claims one for himself while the rest go to Waiter. Proceeding
like this Waiter wins the game if she forces Client to claim all the elements
of any winning set from . In this paper we study fast strategies
for several Waiter-Client games played on the edge set of the complete graph,
i.e. , in which the winning sets are perfect matchings, Hamilton
cycles, pancyclic graphs, fixed spanning trees or factors of a given graph.Comment: 38 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
A Dirac type result on Hamilton cycles in oriented graphs
We show that for each \alpha>0 every sufficiently large oriented graph G with
\delta^+(G),\delta^-(G)\ge 3|G|/8+ \alpha |G| contains a Hamilton cycle. This
gives an approximate solution to a problem of Thomassen. In fact, we prove the
stronger result that G is still Hamiltonian if
\delta(G)+\delta^+(G)+\delta^-(G)\geq 3|G|/2 + \alpha |G|. Up to the term
\alpha |G| this confirms a conjecture of H\"aggkvist. We also prove an Ore-type
theorem for oriented graphs.Comment: Added an Ore-type resul
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