The asymptotic version of the Erd\H{o}s-S\'os conjecture and beyond

Abstract

Klimo\v{s}ov\'a, Piguet, and Rozho\v{n} conjectured that any graph with minimum degree $k/2$ and sufficiently many vertices of degree $k$ should contain all trees with $k$ edges. We prove an asymptotic version of this conjecture for dense host graphs. We obtain interesting corollaries: the first is an asymptotic version of the Erd\H{o}s--S\'os conjecture for dense host graphs, which works without any bounded-degree restriction on the guest trees. Secondly, by leveraging recent results by Pokrovsky, we can translate our results to sparse host graphs in the case of bounded-degree guest trees.