It's similar to Periodicity Lemma. The difference is the edges turn from undirected to directed.

The previos is fake proof.. I can only prove $$$n+m-1$$$ now.

I have completed the proof. Maybe better to say learning the proof from arc127f. Orz maroonrk!

Assume $$$A,B$$$ are coprime. If we can transition from $$$x$$$ to $$$x+A$$$ by $$$+A,-B$$$, add an edge $$$(x,x+A)\bmod B$$$. If $$$n>=A+B$$$, it forms a single loop. We only need to prove the case when $$$n=A+B-1$$$, that there must exists a valid solution.

We can find that all nodes except $$$A$$$ has in-degree, and all except $$$B$$$ has out-degree. So the graph looks like a chain. We can easily construct a topological sequence by the chain.

In fact we have proved the Periodicity Lemma..