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The approximate Loebl-Komlós-Sós Conjecture II: The rough structure of LKS graphs

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    0474809 - MÚ 2018 RIV US eng J - Journal Article
    Hladký, Jan - Komlós, J. - Piguet, Diana - Simonovits, M. - Stein, M. - Szemerédi, E.
    The approximate Loebl-Komlós-Sós Conjecture II: The rough structure of LKS graphs.
    SIAM Journal on Discrete Mathematics. Roč. 31, č. 2 (2017), s. 983-1016. ISSN 0895-4801. E-ISSN 1095-7146
    R&D Projects: GA MŠk(CZ) 1M0545
    EU Projects: European Commission(XE) 628974 - PAECIDM
    Institutional support: RVO:67985840 ; RVO:67985807
    Keywords : extremal graph theory * Loebl–Komlós–Sós conjecture * regularity lemma
    OECD category: Pure mathematics; Pure mathematics (UIVT-O)
    Impact factor: 0.717, year: 2017
    http://epubs.siam.org/doi/10.1137/140982854

    This is the second 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 $(0.5+alpha)n$ vertices of degree at least $(1+alpha)k$ contains each tree $T$ of order $k$ as a subgraph. In the first paper of this 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 third and fourth papers, we refine the structure and use it for embedding the tree $T$.
    Permanent Link: http://hdl.handle.net/11104/0271760

     
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