A Variant of the Erdős-Sós Conjecture

Abstract

A well-known conjecture of Erdős and Sós 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.

References in corpus (4)

• The approximate Loebl-Komlós-Sós Conjecture IV: Embedding techniques and the proof of the main result
• The approximate Loebl-Komlós-Sós Conjecture I: The sparse decomposition
• The approximate Loebl-Komlós-Sós Conjecture II: The rough structure of LKS graphs
• The Approximate Loebl-Komlós-Sós Conjecture III: The finer structure of LKS graphs

Cited by in corpus (1)

• Embedding Nearly Spanning Trees