The approximate Loebl-Komlós--Sós conjecture and embedding trees in sparse graphs

0443855 - MÚ 2016 RIV US eng J - Článek v odborném periodiku
Hladký, Jan - Piguet, Diana - Simonovits, M. - Stein, M. - Szemerédi, E.
The approximate Loebl-Komlós--Sós conjecture and embedding trees in sparse graphs.
Electronic Research Announcements in Mathematical Sciences. Roč. 22, April (2015), s. 1-11. ISSN 1935-9179. E-ISSN 1935-9179
Grant CEP: GA MŠMT(CZ) 1M0545
Institucionální podpora: RVO:67985840 ; RVO:67985807
Klíčová slova: extremal graph theory * Loebl-Komlós-Sós conjecture * regularity lemma
Kód oboru RIV: BA - Obecná matematika
Impakt faktor: 0.333, rok: 2015
http://www.aimsciences.org/journals/displayArticlesnew.jsp?paperID=11110

Loebl, Komlós and Sós conjectured that every n-vertex graph G with at least n/2 vertices of degree at least k contains each tree T of order k+1 as a subgraph. We give a sketch of a proof of the approximate version of this conjecture for large values of k. For our proof, we use a structural decomposition which can be seen as an analogue of Szemerédi's regularity lemma for possibly very sparse graphs. With this tool, each graph can be decomposed into four parts: a set of vertices of huge degree, regular pairs (in the sense of the regularity lemma), and two other objects each exhibiting certain expansion properties. We then exploit the properties of each of the parts of G to embed a given tree T. The purpose of this note is to highlight the key steps of our proof. Details can be found in [arXiv:1211.3050].
Trvalý link: http://hdl.handle.net/11104/0246512