285 research outputs found
Cycle packings in graphs and digraphs
AbstractA cycle packing in a (directed) multigraph is a vertex disjoint collection of (directed) elementary cycles. If D is a demiregular multidigraph we show that the arcs of D can be partitioned into Δin cycle packings — where δin is the maximum indegree of a vertex in D. We then show that the maximum length cycle packings in any digraph contain a common vertex
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
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
Completing Partial Packings of Bipartite Graphs
Given a bipartite graph and an integer , let be the smallest
integer such that, any set of edge disjoint copies of on vertices, can
be extended to an -design on at most vertices. We establish tight
bounds for the growth of as . In particular, we
prove the conjecture of F\"uredi and Lehel \cite{FuLe} that .
This settles a long-standing open problem
Decompositions, Packings, and Coverings of Complete Directed Gaphs with a 3-Circuit and a Pendent Arc.
In the study of Graph theory, there are eight orientations of the complete graph on three vertices with a pendant edge, K3 ∪ {e}. Two of these are the 3-circuit with a pendant arc and the other six are transitive triples with a pendant arc. Necessary and sufficient conditions are given for decompositions, packings, and coverings of the complete digraph with the two 3-circuit with a pendant arc orientations
Packings and Coverings of Various Complete Digraphs with the Orientations of a 4-Cycle.
There are four orientations of cycles on four vertices. Necessary and sufficient conditions are given for covering complete directed digraphs Dv, packing and covering complete bipartite digraphs, Dm,n, and packing and covering the complete digraph on v vertices with hole of size w, D(v,w), with three of the orientations of a 4-cycle, including C4, X, and Y
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