Given a positive integer $n$, let $f$ be a function $$ \{1,\ldots,n\}^2 \to \{1,\ldots,n\}. $$
Then, it is possible that there exists a permutation $\{\sigma_1,\ldots,\sigma_n\}$ of $\{1,\ldots,n\}$ for which $$ \{ f(\sigma_i,\sigma_{i+1}): i=1,\ldots,n\}=\{1,\ldots,n\}, $$ where $\sigma_{n+1}:=\sigma_1$. (In particular, $n$ and $f$ are fixed)
(Rough) Question: Could it be interesting to study the existence of such permutation? Do they have a specific name?
[The problem seems to be vaguely related to Sidon sets: are there any connection?]