For a finite, simple undirected graph $G=(V,E)$ let $\delta(G)$ denote the minimum degree of all vertices. For any integer $k\geq 4$ let $N(k)$ denote the maximum $\delta(G)$ that a connected graph $G$ on $k$ vertices can have such that $G$ does **not** have a Hamiltonian path.

Do we have $\lim \sup_{k\to \infty} \frac{N(k)}{k} = 1$?