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

TL;DR

This paper proves an asymptotic version of a conjecture relating minimum degree conditions to the containment of all trees in dense graphs, extending to sparse graphs for bounded-degree trees.

Contribution

It establishes an asymptotic version of the Erdős–Sós conjecture for dense graphs and extends results to sparse graphs with bounded-degree trees.

Findings

Proves an asymptotic Erdős–Sós conjecture for dense graphs.

Extends results to sparse graphs with bounded-degree trees.

Provides corollaries for trees with no degree restrictions.

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.