A version of the Loebl-Koml\'os-S\'os conjecture for skewed trees

TL;DR

This paper extends the Loebl-Komlós-Sós conjecture to skewed trees, proving an asymptotic version for dense graphs using the regularity method, and deriving bounds on Ramsey numbers for such trees.

Contribution

It introduces a new conjecture for skewed trees and proves its asymptotic validity for dense graphs using advanced combinatorial techniques.

Findings

Asymptotic proof for dense graphs

Bounds on Ramsey numbers for skewed trees

Extension of the Loebl-Komlós-Sós conjecture

Abstract

Loebl, Koml\'os, and S\'os conjectured that any graph with at least half of its vertices of degree at least k contains every tree with at most k edges. We propose a version of this conjecture for skewed trees, i.e., we consider the class of trees with at most k edges such that the sizes of the colour classes of the trees have a given ratio. We show that our conjecture is asymptotically correct for dense graphs. The proof relies on the regularity method. Our result implies bounds on Ramsey number of several trees of given skew.