Given $n$ and $t$ lengths $ l_i, 1\leq l_1<l_2<\cdots<l_t\leq n-1$, of directed diagonals within an $n$-gon such that $l_1+\cdots+l_t\neq 0 \pmod n)$. Does it exist a directed path within the n-gon using each of the given lengths exactly once (and no vertex twice)?
Here the length of a directed diagonal within any kind of an n-gon is the number of sides of the n-gon between the starting and the end vertex counted in a fixed direction. A directed path is a sequence $(d_1,d_2,\ldots,d_t)$ of directed diagonals such that the end vertex of $d_i$ is the starting vertex of $d_{i+1}$ for $i=1,2,…,t-1$.
In number theory, it is equivalent to: For any subset $S=\{l_1,\ldots,l_t\}$ of $\{1,2,…,n-1\}$ with $ l_i, 1\leq l_1<l_2<\cdots<l_t\leq n-1$ and $l_1+\cdots+l_t\neq 0 \pmod n$. Is there any a permutation of the elements of the subset $S$ such that no set of consecutive elements in this permutation has sum $=0 \pmod n$. Can anyone give me a counter-example or prove it?
In Graph theory, I can rephrase this question to: Let $\mathbb{Z}_n$ be a group of integers modulo $n$ and $S=\{l_1,\dots,l_t\}$ be a set of positive integers such that $1\le l_1<l_2< \dots < l_t \le n-1$, where $l_1+\dots + l_t \not\equiv 0 \pmod n$).
Let $P_t$ be a directed path of length $t$ ($t+1$ vertices). Can you label the vertices of the directed path $\overrightarrow{P_t}$ of length $t$ with distinct elements of $\mathbb{Z}_n$ such that the label differences on edges are exactly $S$? The difference is taking by subtract the head by the tail (then take mod $n$).
I have a proof for case $t\leq 6$. However, I wonder is it true or not in general. Many thanks.
