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

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SYSNO ASEP 0474830 J - Journal Article Journal Article Článek ve WOS The Approximate Loebl-Komlos-Sos Conjecture III: The Finer Structure of LKS Graphs Hladký, J. (DE) Komlós, J. (US) Piguet, Diana (UIVT-O) RID, ORCID, SAI Simonovits, M. (HU) Stein, M. (CL) Szemerédi, E. (HU) SIAM Journal on Discrete Mathematics - ISSN 0895-4801 Roč. 31, č. 2 (2017), s. 1017-1071 55 s. eng - English US - United States extremal graph theory ; Loebl–Komlós–Sós conjecture ; regularity lemma BA - General Mathematics Pure mathematics 1M0545 GA MŠk - Ministry of Education, Youth and Sports (MEYS) GJ16-07822Y GA ČR - Czech Science Foundation (CSF) UIVT-O - RVO:67985807 000404770300023 85022094119 10.1137/140982866 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 $T$. Institute of Computer Science Tereza Šírová, sirova@cs.cas.cz, Tel.: 266 053 800 2018
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