Supposing you have a simply connected planar graph of even number of vertices with no loops edges labeled like so:
Kirchoff's laws say that the sum of the edges connected to each vertex is 0 and the sum around each face is 0.
Note there will be no solution if c,g,l, or m in this example is the edge chosen to be 1. But for any the other edges it will either be this solution or -1*all the values.
Then we can plot these solutions...
I don't know how to prove it, it just seems to work in every case. It makes sense to me electrically that if there is a nonzero flow around each face's edges, that some faces will combine and form one flow around the outside. I don't know how to prove that that will always go through every vertex though.