The Loebl-Komlos-Sos conjecture for trees of diameter 5 and for certain caterpillars

TL;DR

This paper proves the Loebl-Komlos-Sos conjecture for trees with diameter at most 5 and certain caterpillars, establishing new bounds on related Ramsey numbers.

Contribution

It extends the conjecture's validity to specific tree classes and derives bounds on their Ramsey numbers, advancing understanding in graph theory.

Findings

Proved the conjecture for trees of diameter ≤ 5.

Validated the conjecture for a class of caterpillars.

Derived bounds on Ramsey numbers for these trees.

Abstract

Loebl, Komlos, and Sos conjectured that if at least half the vertices of a graph G have degree at least some k, then every tree with at most k edges is a subgraph of G. We prove the conjecture for all trees of diameter at most 5 and for a class of caterpillars. Our result implies a bound on the Ramsey number r(T,F) of trees T, F from the above classes.