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The approximate Loebl-Komlós-Sós Conjecture I: The sparse decomposition
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SYSNO ASEP 0474810 Document Type J - Journal Article R&D Document Type Journal Article Subsidiary J Článek ve WOS Title The approximate Loebl-Komlós-Sós Conjecture I: The sparse decomposition Author(s) Hladký, Jan (MU-W) RID, SAI, ORCID
Komlós, J. (US)
Piguet, Diana (UIVT-O) RID, ORCID, SAI
Simonovits, M. (HU)
Stein, M. (CL)
Szemerédi, E. (HU)Source Title SIAM Journal on Discrete Mathematics. - : SIAM Society for Industrial and Applied Mathematics - ISSN 0895-4801
Roč. 31, č. 2 (2017), s. 945-982Number of pages 38 s. Language eng - English Country US - United States Keywords extremal graph theory ; Loebl–Komlós–Sós conjecture ; regularity lemma Subject RIV BA - General Mathematics OECD category Pure mathematics Subject RIV - cooperation Institute of Computer Science - General Mathematics R&D Projects 1M0545 GA MŠMT - Ministry of Education, Youth and Sports (MEYS) Institutional support MU-W - RVO:67985840 ; UIVT-O - RVO:67985807 UT WOS 000404770300021 EID SCOPUS 85021932060 DOI 10.1137/140982842 Annotation In a series of four papers 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 $(0.5+alpha)n$ vertices of degree at least $(1+alpha)k$ 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. Workplace Mathematical Institute Contact Jarmila Štruncová, struncova@math.cas.cz, library@math.cas.cz, Tel.: 222 090 757 Year of Publishing 2018
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