  In this work we consider a generalization of graph flows. A graph flow is, in
its simplest formulation, a labeling of the directed edges with real numbers
subject to various constraints. A common constraint is conservation in a
vertex, meaning that the sum of the labels on the incoming edges of this vertex
equals the sum of those on the outgoing edges. One easy fact is that if a flow
is conserving in all but one vertex, then it is also conserving in the
remaining one. In our generalization we do not label the edges with real
numbers, but with elements from an arbitrary group, where this fact becomes
false in general. As we will show, graphs with the property that conservation
of a flow in all but one vertex implies conservation in all vertices are
precisely the planar graphs.
