About Counterexamples to The 5-Flow Conjecture

Let C be an edge cut on a graph G. We study correlations among numbers of nowhere-zero flows with prescribed values on C. They can be characterized by linear algebra. We apply this approach to obtain restrictions for the smallest counterexamples to the 5-flow conjecture of Tutte (every bridgeless graph has a nowhere-zero 5-flow). In particular, we show that this counterexample must be cyclically 6-edge connected and has girth at least 9.