The large first- and second-neighbourhood conjecture for tree embeddings

Let $k\in\mathbb{N}$, and let $G$ be a graph. For a vertex $x\in V(G)$, let $N(x)$ denote its neighbourhood and let $N_2(x)$ denote its second neighbourhood. Large first- and second-neighbourhood conjecture. If

$$\delta(G)\geq \frac{k}{2}$$

and there is a vertex $x$ such that

$$\min\{|N(x)|,|N_2(x)|\}\geq \frac{4k}{3},$$

then $G$ contains every tree with $k$ edges.

This offers a local alternative to imposing a large maximum degree, by requiring both the first and second neighbourhoods of one vertex to be large. The source presents it as an open approach to avoiding known extremal obstructions; no resolution is supplied.