Embedding large subgraphs into dense graphs

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.

Local resilience of spanning subgraphs in sparse random graphs

“…There is a series of other well-known results that can be restated in terms of local resilience of complete graphs with respect to containing spanning subgraphs with bounded maximum degree, such as powers of Hamilton cycles, trees, clique-factors, and H-factors (see e.g. [12] for a survey). Schacht and two of the current authors [6] extended these results to families of graphs with sublinear bandwidth, where a graph is defined to have bandwidth at most b if there is a labelling of its vertex set by integers 1, .…”

Fractional and integer matchings in uniform hypergraphs

“…Given a k-uniform hypergraph G = (V, E) and S ∈ For integers n, k, d, s satisfying 0 ≤ d ≤ k − 1 and 0 ≤ s ≤ n/k, we let m s d (k, n) denote the minimum integer m such that every k-uniform hypergraph G on n vertices with δ d (G) ≥ m has a matching of size s. So the results discussed in Section 1.1 correspond to the case d = 0. The following degree condition for forcing perfect matchings has been conjectured in [13,18] and also received much attention recently. …”

Finding Hamilton cycles in robustly expanding digraphs

“…For surveys on applications of the Regularity Lemma and the Blow-up Lemma, we refer the reader to [19,16,20].…”

“…(See e.g. [20] for a sketch proof.) We will also use the Diregularity Lemma in its algorithmic form.…”

“…This is obtained in a similar way from the standard version as in the undirected case: one applies the Diregularity lemma with a parameter ε ε and then deletes a small proportion of the edges (in particular all edges between pairs which are not ε -regular or have density less than d + ε ) and moves a small proportion of the vertices into V 0 . (See [20] for a sketch of this for the undirected case.) One important difference is that in our case we do not know whether each pair is ε -regular or not.…”