Abstract

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 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.

Introduction, overview and basic notation

In this survey, we study the question of when a graph G contains some given large or spanning graph H as a subgraph. Many important problems can be phrased in this way: one example is Dirac's theorem, which states that every graph G on n ≥ 3 vertices with minimum degree at least n/2 contains a Hamilton cycle.