Embedding large subgraphs into dense graphs

TL;DR

This paper surveys recent advances in embedding large subgraphs into dense graphs, focusing on conditions like minimum degree that guarantee perfect packings, Hamiltonicity, and tree embeddings, using tools like the Regularity and Blow-up lemmas.

Contribution

It provides an overview of recent progress and methods in embedding large subgraphs into dense graphs, emphasizing F-packings, Hamiltonicity, and tree embeddings.

Findings

Progress on conditions guaranteeing perfect F-packings

Applications of Regularity and Blow-up lemmas in graph embedding

Resolution of several longstanding problems and conjectures

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 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.