The Approximate Loebl – Komlós – Sós Conjecture and Embedding Trees in Sparse Graphs

@inproceedings{Hladk2014TheAL,
  title={The Approximate Loebl – Koml{\'o}s – S{\'o}s Conjecture and Embedding Trees in Sparse Graphs},
  author={Jan Hladk{\'y} and Diana Piguet and Mikl{\'o}s Simonovits and Maya Jakobine Stein and Endre Szemer{\'e}di and Loebl},
  year={2014}
}
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… CONTINUE READING
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