A sequence with the property that it can be be split into n pieces where each piece sums to x, and no other such split exists

Question

I am looking for a set of sequences (specifically cyclic graphs, see property 5) that adhere to these properties:

1. They are of length $m*n$
2. They can be split into $n$ segments each of length $m$
3. Each integer in the sequence is a member of some composition of the integer $x$.
4. There is a unique split into $n$ chunks of length $m$ such that all chunks add up to $x$.
5. These chunks can be wrapped around, that is the sequence can be viewed as a cycle graph where each node is one of the members of the sequence and a split should remove edges such that there are $n$ path graphs(chunks) each containing $m$ vertices.

If $m$ is 2 there will obviously be 2 possible splits, only one of them should exhibit this behavior if the sequence is a member of this set of sequences. If $m$ is 3 then there will be 3 possible splits, only one should exhibit the properties above. etc.