The approximate Loebl-Koml\'os-S\'os Conjecture II: The rough structure of LKS graphs

TL;DR

This paper advances the understanding of the Loebl-Komlos-Sos Conjecture by identifying a key combinatorial structure within graph decompositions, facilitating the embedding of trees in graphs with certain degree conditions.

Contribution

It introduces a new combinatorial structure inside a graph decomposition that aids in embedding trees, extending previous partial results on the conjecture.

Findings

Decomposition of graphs into parts with distinct characteristics

Identification of a combinatorial structure within the decomposition

Foundation for embedding trees in graphs with degree constraints

Abstract

This is the second 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; this decomposition might be viewed as an analogue of a regular partition for sparse graphs. In the present paper, we find a combinatorial structure inside this decomposition. In the last two papers, we refine the structure and use it for embedding the tree $T$.