Six-flows of signed graphs with frustration index three

Bouchet's 6-flow conjecture states that every flow-admissible signed graph admits a nowhere-zero 6-flow. Seymour's 6-flow theorem states that the conjecture holds for balanced signed graphs. Rollová et al. show that every flow-admissible signed graph with frustration index two admits a nowhere-zero 7-flow, where the frustration index of a signed graph is the smallest number of edges whose deletion leaves a balanced signed graph. Wang et al. improve the result to 6-flows. In this paper, we further extend these results, and confirm Bouchet's 6-flow conjecture for signed graphs with frustration index three. There are infinitely many signed graphs with frustration index three admitting a nowhere-zero 6-flow but no 5-flow. From the point of view of flow theory, signed graphs with frustration index two are very similar to those of ordinary graphs. However, there are significant differences between ordinary graphs and signed graphs with frustration index three.