The approximate Loebl-Koml\'os-S\'os Conjecture

TL;DR

This paper proves a version of the Loebl-Komlos-Sos Conjecture, showing that large graphs with certain degree conditions contain all trees of a given size, using a novel decomposition approach beyond the Regularity Lemma.

Contribution

It introduces a new decomposition technique that extends the Regularity Lemma to sparse graphs, enabling the proof of the conjecture for a broader class of graphs.

Findings

Graphs with high minimum degree contain all large trees of a certain size.

The new decomposition method works for sparse graphs where traditional methods fail.

The result confirms the conjecture for sufficiently large graphs under specified degree conditions.

Abstract

We prove the following version of the Loebl-Komlos-Sos Conjecture: For every alpha>0 there exists a number M such that for every k>M 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. The method to prove our result follows a strategy common to approaches which employ the Szemeredi Regularity Lemma: we decompose the graph G, find a suitable combinatorial structure inside the decomposition, and then embed the tree T into G using this structure. However, the decomposition given by the Regularity Lemma is not of help when G is sparse. To surmount this shortcoming we use a more general decomposition technique: each graph can be decomposed into vertices of huge degree, regular pairs (in the sense of the Regularity Lemma), and two other objects each exhibiting certain expansion properties.