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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 Szemerédi and the Blow-up lemma of Komlós, Sárközy and Szemerédi 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.

Citation Context ...n − i, (ii) d − i ≥ i + 1 or d+ n−i ≥ n − i. It is even an open problem whether the conditions imply the existence of a cycle through any pair of given vertices (see =-=[12]-=-).

There are several survey articles =-=[20, 21, 47]-=- (the second one contains results on general digraphs too) and a book by J. The most famous examples of results of this kind are probably Dirac’s theorem on Hamilton cycles and Tutte’s theorem on perfect matchings.

Moon [58] where the properties of tournaments are considered. 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.

We describe results (theorems and algorithms) on directed walks in semicomplete m- partite digraphs including s ..." A digraph obtained by replacing each edge of a complete m-partite graph by an arc or a pair of mutually opposite arcs with the same end vertices is called a semicomplete m-partite digraph. What conditions ensure that a graph G contains some given spanning subgraph H?

We describe results (theorems and algorithms) on directed walks in semicomplete m- partite digraphs including some recent results concerning tournaments. The most famous examples of results of this kind are probably Dirac’s theorem on Hamilton cycles and Tutte’s theorem on perfect matchings.

For completeness of the presentation of both particular questions and the general area, it also contains material on closely related topics such as traceable, pancyclic and hamiltonian-connected graphs and digraphs. A digraph obtained by replacing each edge of a complete m-partite graph by an arc or a pair of mutually opposite arcs with the same end vertices is called a semicomplete m-partite digraph.

It is easy to see that one cannot omit the condition that G is strongly connected.

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