Signed graphs with two negative edges

The presented paper studies the flow number $F(G,\sigma )$ of flow-admissible signed graphs $(G,\sigma )$ with two negative edges. We restrict our study to cubic graphs, because for each non-cubic signed graph $(G,\sigma )$ there is a set $\mathcal{G}(G,\sigma )$ of cubic graphs such that $F(G, \sigma) \leq \min {F(H,\sigma_H) : (H,\sigma_H) \in {\cal G}(G)}$\mathrm{.}Weprovethat$F(G,\sigma) \leq 6$if$(G,\sigma)$containsabridgeand$F(G,\sigma) \leq 7$ in general. We prove better bounds, if there is an element $(H,{\sigma }_H)$ of ${\cal G}(G,\sigma)$whichsatisfiessomeadditionalconditions\mathrm{.}Inparticular,if$H$isbipartite,then$F(G,\sigma) \leq 4$andtheboundistight\mathrm{.}If$H$ is 3-edge-colorable or critical or if it has a sufficient cyclic edge-connectivity, then $F(G,\sigma )\le 6$. Furthermore, if Tutte’s 5-Flow Conjecture is true, then $(G,\sigma )$ admits a nowhere-zero 6-flow endowed with some strong properties.