Broersma's degree condition conjecture for hamiltonian claw-free graphs

Let $G$ be a 2-connected claw-free graph on $n$ vertices. An end-vertex of an induced copy of $N$ in $G$ is a vertex serving as an end-vertex in that copy.

Broersma's conjecture. If every end-vertex of an induced copy of $N$ in $G$ has degree at least $(n-2)/3$, then $G$ is hamiltonian.

This conjecture concerns degree conditions guaranteeing hamiltonicity in 2-connected claw-free graphs. The paper notes that it remains open; its results establish the analogous assertion with the stronger bound $|V(G)|/3+1$, giving an affirmative solution up to an additive constant.