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arXiv · 1408.3870

The approximate Loebl-Koml\'os-S\'os Conjecture IV: Embedding techniques and the proof of the main result

Jan Hladký·János Komlós·Diana Piguet·Miklós Simonovits·Maya J. Stein·Endre Szemerédi
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Abstract

This is the last paper of a series of four papers in which we prove the following relaxation of the Loebl-Komlos-Sos Conjecture: For every $\alpha>0$ there exists a number~$k_0$ such that for every $k>k_0$ every $n$-vertex graph $G$ with at least $(\frac12+\alpha)n$ vertices of degree at least $(1+\alpha)k$ 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-Komlos-Sos Conjecture contains one of ten specific configurations. In this paper we embed the tree $T$ in each of the ten configurations.

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Jan Hladký, János Komlós, Diana Piguet, Miklós Simonovits, Maya J. Stein, Endre Szemerédi. 2014-08-17. The approximate Loebl-Koml\'os-S\'os Conjecture IV: Embedding techniques and the proof of the main result. https://doi.org/10.1137/140982878

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