# Is a network flow equivalent to a walk?

Consider a directed graph where each edge has infinite capacity.
Now consider a directed walk on this graph from arbirtary node s to arbitrary node t with $$s \neq t$$.

Definition (Directed walk): A walk is a sequence of not necessarily distinct edges directed in the same direction which joins a sequence of vertices. It differs from a path because a walk can contain cycles.

With this walk we can build an s,t-flow of size 1 where the following condition holds: the flow on every edge e is equal to how manny times e appears in the walk.

I'm pretty sure the flow we just built is valid because if the walk passes i times from node v with $$v \notin$$ {$$s,t$$} then node v sees a flow of size i entering and a flow of size i leaving. It's not a formal proof but I think I can write one down.

What I'm not so sure about is if we can do exactly the reverse: build a directed walk between nodes s and t for every s,t-flow of size 1 such that the same condition as above holds.

My question is: is it possible to build a walk for every flow, and if yes how would I go about prooving it?

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Consider the sum of a flow of $$1$$ along a directed path from $$s$$ to $$t$$ and a flow of $$1$$ along a directed cycle that does not share any nodes with the path.