The Approximate Loebl-Koml\'os-S\'os Conjecture III: The finer structure of LKS graphs

TL;DR

This paper refines the structural understanding of graphs satisfying a relaxed degree condition related to the Loebl-Komlos-Sos Conjecture, enabling better embedding of trees of a given size.

Contribution

It provides a refined combinatorial structure within the graph decomposition, advancing the approach to embedding trees in graphs meeting the conjecture's relaxed conditions.

Findings

Decomposition of graphs into parts with specific properties

Identification of a refined combinatorial structure

Preparation for embedding trees in the final paper

Abstract

This is the third 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 paper of the series, we gave a decomposition of the graph $G$ into several parts of different characteristics. In the second paper, we found a combinatorial structure inside the decomposition. In this paper, we will give a refinement of this structure. In the forthcoming fourth paper, the refined structure will be used for embedding the tree $T$.