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arXiv · 1408.3871

The approximate Loebl-Koml\'os-S\'os Conjecture II: The rough structure of LKS graphs

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

This is the second 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 paper of the series, we gave a decomposition of the graph $G$ into several parts of different characteristics; this decomposition might be viewed as an analogue of a regular partition for sparse graphs. In the present paper, we find a combinatorial structure inside this decomposition. In the last two papers, we refine the structure and use it for embedding the tree $T$.

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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 II: The rough structure of LKS graphs. https://doi.org/10.1137/140982854

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