Havet–Reed–Wood–Stein tree-containment conjecture

Prove that every graph with minimum degree at least \(\lfloor 2k/3\rfloor\) and maximum degree at least \(k\) contains every tree with \(k\) edges.

Background

The paper studies degree conditions guaranteeing that a host graph or digraph contains a prescribed tree. The authors present the Havet–Reed–Wood–Stein conjecture as a motivation for seeking analogous results for smaller trees and weaker minimum-degree assumptions. The conjecture asserts that a minimum-degree threshold of approximately $2k/3$, together with one vertex of degree at least $k$, suffices to contain every $k$-edge tree.

The paper notes that the minimum-degree bound is best possible, using a construction consisting of two cliques of order approximately $2k/3$ joined through a universal vertex. The conjecture itself is not proved in the paper; instead, the paper establishes an approximate bounded-degree analogue in a related setting.