On November 7, 2023, Bruce A. Reed from the Academia Sinica, Taiwan gave a talk at the Discrete Math Seminar on the existence of every k-vertex tree as a minor in a graph with certain lower bounds on the minimum degree and the maximum degree. The title of his talk was “Some Variants of the Erdős-Sós Conjecture“.
Determining the density required to ensure that a host graph G contains some target graph as a subgraph or minor is a natural and well-studied question in extremal combinatorics. The celebrated 50-year-old Erdős-Sós conjecture states that for every k, if G has average degree exceeding k-2 then it contains every tree T with k vertices as a subgraph. This is tight as the clique with k-1 vertices contains no tree with k vertices as a subgraph.
We present some variants of this conjecture. We first consider replacing bounds on the average degree by bounds on the minimum and maximum degrees. We then consider replacing subgraph by minor in the statement.