### Abstract

In a series of four papers 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. The method to prove our result follows a strategy similar to approaches that employ the Szemerédi regularity lemma: We decompose the graph G, find a suitable combinatorial structure inside the decomposition, and then embed the tree T into G using this structure. Since for sparse graphs G, the decomposition given by the regularity lemma is not helpful, we use a more general decomposition technique. We show that each graph can be decomposed into vertices of huge degree, regular pairs (in the sense of the regularity lemma), and two other objects each exhibiting certain expansion properties. In this paper, we introduce this novel decomposition technique. In the three follow-up papers, we find a suitable combinatorial structure inside the decomposition, which we then use for embedding the tree.

Original language | English |
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Pages (from-to) | 945-982 |

Number of pages | 38 |

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)

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## Cite this

*SIAM Journal on Discrete Mathematics*,

*31*(2), 945-982. https://doi.org/10.1137/140982842