Circular flow on signed graphs

The circular flow number Φ c ( G , σ ) of a signed graph ( G , σ ) is the minimum r for which an orientation of ( G , σ ) admits a circular r-flow. We prove that the circular flow number of a signed graph ( G , σ ) is equal to the minimum imbalance ratio of an orientation of ( G , σ ) . We then use this result to prove that if G is 4-edge-connected and ( G , σ ) has a nowhere zero flow, then Φ c ( G , σ ) (as well as Φ ( G , σ ) ) is at most 4. If G is 6-edge-connected and ( G , σ ) has a nowhere zero flow, then Φ c ( G , σ ) is strictly less than 4.