I've stumbled across a short note from 1993 where a nice question was asked: Suppose you take a graph with vertices $\{1,2,\ldots,n\}$ and connect $i,j$ by an edge if and only if $i+j$ is a $k$th power of some number. Call this graph $G_{k}(n)$. The authors of the note found that $G_{2}(32)$ is Hamiltonian (and that $32$ is the first $n$ for which $G_{2}(n)$ is Hamiltonian). They conjectured that $G_{2}(n)$ is Hamiltonian for all $n \geq 32$.

I found no citations of this paper so I wonder if someone has attacked this question since.

UPDT: There is some discussion of the $k=3$ case here, from an "elementary" point of view.