Nowhere-zero 3-flows in signed planar graphs (Q6574386)

In the field of planar graph coloring, Grötzsch's 3-coloring theorem is an influential result. It states that every triangle free planar graph has a proper 3-coloring. As a generalization of the dual concept of graph coloring, \textit{W. T. Tutte} [Proc. Lond. Math. Soc. (2) 51, 474--483 (1949; Zbl 0033.30803)] initiated the study of flow theory and observed that a plane graph admits a nowhere-zero \(k\)-flow if and only if its dual graph has a proper \(k\)-coloring. Motivated by the 3-coloring theorem of Grötzsch, Tutte proposed his 3-flow conjecture namely: Every 4-edge-connected graph admits a nowhere-zero 3-flow. As a major open problem in the flow theory, Tutte's 3-flow conjecture is still open as of today. The authors here generalize Grötzsch's theorem to signed planar graphs by showing that every 4-edge-connected signed planar graph with two negative edges admits a 3-NZF (nowhere zero flow). Their proof employs the flow extension ideas from \textit{R. Steinberg} and \textit{D. H. Younger} [Ars Comb. 28, 15--31 (1989; Zbl 0721.05025)] and \textit{C. Thomassen} [J. Comb. Theory, Ser. B 62, No. 2, 268--279 (1994; Zbl 0817.05024); ibid. 88, No. 1, 189--192 (2003; Zbl 1025.05022)], as well as refined exploration of the location of negative edges and elaborated discharging arguments in signed planar graphs.