The approximate Loebl-Komlós-Sós Conjecture II: The rough structure of LKS graphs

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 $(\frac{1}{2}+\alpha )n$ 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 $T$.

38 pages, 4 figures; new is Section 5.1.1; accepted to SIDMA

References in corpus (4)

• The approximate Loebl-Komlós-Sós Conjecture IV: Embedding techniques and the proof of the main result
• The approximate Loebl-Komlós-Sós Conjecture I: The sparse decomposition
• The approximate Loebl-Komlós-Sós Conjecture II: The rough structure of LKS graphs
• The Approximate Loebl-Komlós-Sós Conjecture III: The finer structure of LKS graphs

Cited by in corpus (6)

• The approximate Loebl-Komlós-Sós Conjecture IV: Embedding techniques and the proof of the main result
• A Variant of the Erdős-Sós Conjecture
• The approximate Loebl-Komlós-Sós Conjecture II: The rough structure of LKS graphs
• The approximate Loebl-Komlós-Sós Conjecture I: The sparse decomposition
• The Approximate Loebl-Komlós-Sós Conjecture III: The finer structure of LKS graphs
• The approximate Loebl-Komlos-Sos conjecture and embedding trees in sparse graphs