4 problems

Faudree–Schelp conjecture. Every such pair $u,v$ has a path of every length $k$ in this range. The conjecture was disproved by Thomassen, who constructed hamiltonian-connected grap…

A 1–3 tree is a tree in which every vertex has degree either $1$ or $3$. For a tree, a leaf-to-leaf path is a path whose endpoints are leaves. Narins–Pokrovskiy–Szabó conjecture. T…

A 1–3 tree is a tree in which every vertex has degree either $1$ or $3$. Narins–Pokrovskiy–Szabó conjecture. Every 1–3 tree $T$ of order $n$ has leaf-to-leaf paths of at least … di…

A 1-3 tree is a tree whose vertices have degrees $1$ or $3$, and a leaf-leaf path is a path with leaf endpoints. Many-short-paths conjecture. There is a constant $\alpha>0$ and a f…