The approximate Loebl-Koml\'os-S\'os Conjecture IV: Embedding techniques and the proof of the main result

TL;DR

This paper completes a series proving a relaxed version of the Loebl-Komlos-Sos Conjecture by embedding trees into specific graph configurations using advanced decomposition and embedding techniques.

Contribution

It introduces a novel embedding approach within ten configurations, finalizing the proof of the approximate conjecture for large graphs.

Findings

Successfully embedded trees in all ten configurations.

Confirmed the conjecture holds for graphs with high minimum degree.

Provided a comprehensive framework for tree embedding in dense graphs.

Abstract

This is the last paper of a series of four papers in which we prove the following relaxation of the Loebl-Komlos-Sos 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 $(\frac12+\alpha)n$ vertices of degree at least $(1+\alpha)k$ contains each tree $T$ of order $k$ as a subgraph. In the first two papers of this series, we decomposed the host graph $G$, and found a suitable combinatorial structure inside the decomposition. In the third paper, we refined this structure, and proved that any graph satisfying the conditions of the above approximate version of the Loebl-Komlos-Sos Conjecture contains one of ten specific configurations. In this paper we embed the tree $T$ in each of the ten configurations.