### Abstract

This is the last of a series of four papers in which we prove the following relaxation of the Loebl-Komlós-Sós conjecture: For every α > 0 there exists a number k_{0} such that for every k > k_{0}, every n-vertex graph G with at least (1/2 + α)n vertices of degree at least (1 + α)k contains each tree T of order k as a subgraph. In the first two papers of this series, we decomposed the host graph G and found a suitable combinatorial structure inside the decomposition. In the third paper, we refined this structure and proved that any graph satisfying the conditions of the above approximate version of the Loebl-Komlós-Sós conjecture contains one of ten specific configurations. In this paper we embed the tree T in each of the ten configurations.

Original language | English |
---|---|

Pages (from-to) | 1072-1148 |

Number of pages | 77 |

Journal | SIAM Journal on Discrete Mathematics |

Volume | 31 |

Issue number | 2 |

DOIs | |

Publication status | Published - Jan 1 2017 |

### Keywords

- Extremal graph theory
- Graph decomposition
- Loebl-Komlós-Sós conjecture
- Regularity lemma
- Sparse graph
- Tree embedding

### ASJC Scopus subject areas

- Mathematics(all)

## Fingerprint Dive into the research topics of 'The approximate Loebl-Komlós-Sós conjecture IV: Embedding techniques and the proof of the main result'. Together they form a unique fingerprint.

## Cite this

*SIAM Journal on Discrete Mathematics*,

*31*(2), 1072-1148. https://doi.org/10.1137/140982878