A Variant of the Erd\H{o}s-S\'os Conjecture

TL;DR

This paper proposes a new variant of the Erd ext{"o}s-S ext{"o}s conjecture relating graph degree conditions to the containment of all trees with a given number of edges, supported by partial results.

Contribution

It introduces a modified conjecture involving maximum and minimum degree conditions and provides partial evidence for its validity.

Findings

Existence of a function g(m) satisfying a weakened conjecture

Proof of a positive gamma for a further weakened conjecture

Partial validation of the variant through specific degree conditions

Abstract

A well-known conjecture of Erd\H{o}s and S\'os states that every graph with average degree exceeding $m-1$ contains every tree with $m$ edges as a subgraph. We propose a variant of this conjecture, which states that every graph of maximum degree exceeding $m$ and minimum degree at least $\lfloor \frac{2m}{3}\rfloor$ contains every tree with $m$ edges. As evidence for our conjecture we show (i) for every $m$ there is a $g(m)$ such that the weakening of the conjecture obtained by replacing $m$ by $g(m)$ holds, and (ii) there is a $\gamma>0$ such that the weakening of the conjecture obtained by replacing $\lfloor \frac{2m}{3}\rfloor$ by $(1-\gamma)m$ holds.