The approximate Loebl-Komlós-Sós Conjecture II: The rough structure of LKS graphs
Result description
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 $T$.
Keywords
extremal graph theoryLoebl–Komlós–Sós conjectureregularity lemma
The result's identifiers
Result code in IS VaVaI
Alternative codes found
RIV/67985840:_____/17:00474809
Result on the web
DOI - Digital Object Identifier
Alternative languages
Result language
angličtina
Original language name
The approximate Loebl-Komlós-Sós Conjecture II: The rough structure of LKS graphs
Original language description
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 $T$.
Czech name
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Czech description
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Classification
Type
Jimp - Article in a specialist periodical, which is included in the Web of Science database
CEP classification
—
OECD FORD branch
10101 - Pure mathematics
Result continuities
Project
Continuities
I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace
Others
Publication year
2017
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů
Data specific for result type
Name of the periodical
SIAM Journal on Discrete Mathematics
ISSN
0895-4801
e-ISSN
—
Volume of the periodical
31
Issue of the periodical within the volume
2
Country of publishing house
US - UNITED STATES
Number of pages
34
Pages from-to
983-1016
UT code for WoS article
000404770300022
EID of the result in the Scopus database
2-s2.0-85021890019
Basic information
Result type
Jimp - Article in a specialist periodical, which is included in the Web of Science database
OECD FORD
Pure mathematics
Year of implementation
2017