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# The Approximate Loebl-Komlos-Sos Conjecture III: The Finer Structure of LKS Graphs

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0474830 - ÚI 2018 RIV US eng J - Journal Article
Hladký, J. - Komlós, J. - Piguet, Diana - Simonovits, M. - Stein, M. - Szemerédi, E.
The Approximate Loebl-Komlos-Sos Conjecture III: The Finer Structure of LKS Graphs.
SIAM Journal on Discrete Mathematics. Roč. 31, č. 2 (2017), s. 1017-1071. ISSN 0895-4801. E-ISSN 1095-7146
R&D Projects: GA MŠk(CZ) 1M0545; GA ČR GJ16-07822Y
Grant - others: EPRSC(GB) EP/D063191/1; EPRSC(GB) EP/J501414/1; FP7(XE) PIEF-GA-2009-253925; GA MŠK(CZ) CZ.1.05/1.1.00/02.0090
Institutional support: RVO:67985807
Keywords : extremal graph theory * Loebl–Komlós–Sós conjecture * regularity lemma
OECD category: Pure mathematics
Impact factor: 0.717, year: 2017

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 $\alpha>0$ 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. 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 $T$.