Abstract

A well-known conjecture of Erdős and Sós states that every graph with average degree exceeding (Formula presented.) contains every tree with (Formula presented.) edges as a subgraph. We propose a variant of this conjecture, which states that every graph of maximum degree exceeding (Formula presented.) and minimum degree at least (Formula presented.) contains every tree with (Formula presented.) edges. As evidence for our conjecture we show (a) for every (Formula presented.) there is a (Formula presented.) such that the weakening of the conjecture obtained by replacing the first (Formula presented.) by (Formula presented.) holds, and (b) there is a (Formula presented.) such that the weakening of the conjecture obtained by replacing (Formula presented.) by (Formula presented.) holds.