Six-flows of signed graphs with frustration index three (Q6646437)

A signed graph \((G,\sigma)\) is a graph \(G\) associated with a signature \(\sigma:E(G)\to\{\pm 1\}\). \textit{W. T. Tutte} [Can. J. Math. 6, 80--91 (1954; Zbl 0055.17101); Proc. Lond. Math. Soc. (2) 51, 474--483 (1949; Zbl 0033.30803)] formulated the theory of integer flows, a dual concept to the vertex coloring of planar graphs. His flow theory is extended to signed graphs by the authors here. It extends naturally from the probe of graphs embedded on nonorientable surfaces, where nowhere-zero flow emerges as the dual of local tension. The authors attempt to settle a conjecture raised by \textit{A. Bouchet} [J. Comb. Theory, Ser. B 34, 279--292 (1983; Zbl 0518.05058)] namely every flow-admissible signed graph admits a nowhere-zero 6-flow and confirm it for signed graphs with frustration index three.