A variant of the Erdős‐Sós conjecture

International audience; 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 $[\frac{2m}{3}]$ 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 the first m by g(m) holds, and (ii) there is a $\gamma > 0$ such that the weakening of the conjecture obtained by replacing $[\frac{2m} {3}$ by $(1 − \gamma)m$ holds.