Let G be a graph with oriented edges. One can assign to each edge of G an integer weight. This assignment is a flow if it conserves matter at each vertex, i.e. the sum of weights on the vertex's inward edges is equal to the sum on its outward edges. Can one assign such a flow to G without using 0 as a weight? Collecting all graphs for which such a non-zero flow exists, can you bound the absolute value of the weights you use? What is the best possible bound?
In this talk we are going to (partially) answer these questions, connect the non-zero flows to the four-color theorem and discuss some open problems.