### Abstract

This is the third 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 paper of the series, we gave a decomposition of the graph G into several parts of different characteristics. In the second paper, we found a combinatorial structure inside the decomposition. In this paper, we will give a refinement of this structure. In the fourth paper, the refined structure will be used for embedding the tree T.

Original language | English (US) |
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Pages (from-to) | 1017-1071 |

Number of pages | 55 |

Journal | SIAM Journal on Discrete Mathematics |

Volume | 31 |

Issue number | 2 |

DOIs | |

State | Published - 2017 |

### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Keywords

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

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## Cite this

*SIAM Journal on Discrete Mathematics*,

*31*(2), 1017-1071. https://doi.org/10.1137/140982866