**Motivation.** In a team of $n$ people, we had the task to build subteams of a fixed size $k<n$ such that every day, $1$ person of the subteam is replaced by another person in the team, but not in the subteam (so that the new subteam consisted of $k$ again). The question arose if and for what choices of $k$ and $n$ the subteam schedule can be built to contain each choice of $k$ people out of the team exactly once.

**Formal version.** For any set $X$ and positive integer $k$, let $[X]^k$ be the collection of subset of $X$ having $k$ elements. For $n\in\mathbb{N}$ let $[n] =\{1,\ldots,n\}$. For integers $1< k < n$ we define a graph $G(n, k)$ by $V(G(n,k)) = [[n]]^k$ and $$E(G(n,k)) = \big\{\{A, B\} : A, B \in [[n]]^k \land |A\cap B| = k-1\big\}.$$
For what choices of $1< k < n$ does $G(n,k)$ have a Hamiltonian path? Bonus question: Replace "path" by "cycle" in the original question. (The bonus question need not be answered for acceptance.)

Art Of Computer Programming, on combinatorial algorithms. $\endgroup$ – Steven Stadnicki Jun 17 '20 at 20:32