Degree conditions for embedding trees

TL;DR

This paper proposes degree conditions for embedding all trees with a given number of edges into dense graphs, providing approximate solutions for bounded degree trees and implications for related conjectures.

Contribution

It introduces an approximate degree condition conjecture for embedding trees and proves it for bounded degree trees in dense graphs, advancing understanding of related conjectures.

Findings

Proved an approximate version of the conjecture for bounded degree trees.

Established implications for the Erdős–Sós conjecture.

Extended results to the 2/3-conjecture.

Abstract

We conjecture that every $n$-vertex graph of minimum degree at least $\frac k2$ and maximum degree at least $2k$ contains all trees with $k$ edges as subgraphs. We prove an approximate version of this conjecture for trees of bounded degree and dense host graphs. Our work also has implications on the Erd\H os--S\'os conjecture and the $\frac 23$-conjecture. We prove an approximate version of both conjectures for bounded degree trees and dense host graphs.