The approximate Loebl-Komlos-Sos conjecture and embedding trees in sparse graphs

TL;DR

This paper presents an approximate proof of the Loebl-Komlós-Sós conjecture for large trees, using a novel structural decomposition akin to Szemerédi's regularity lemma tailored for sparse graphs.

Contribution

It introduces a new decomposition method for sparse graphs and demonstrates how to embed large trees, advancing understanding of the conjecture in sparse graph settings.

Findings

Established an approximate version of the conjecture for large k

Developed a structural decomposition for sparse graphs

Enabled tree embedding using expansion and regularity properties

Abstract

Loebl, Koml\'os and S\'os 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 Szemer\'edi'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].