Abstract
General results on nowhere-zero integral chain groups are proved and then specialized to the case of flows in bidirected graphs. For instance, it is proved that every 4-connected (resp. 3-connected and balanced triangle free) bidirected graph which has at least an unbalanced circuit and a nowhere-zero flow can be provided with a nowhere-zero integral flow with absolute values less than 18 (resp. 30). This improves, for these classes of graphs, Bouchet's 216-flow theorem ( J. Combin. Theory Ser. B 34 (1982), 279–292). We also approach his 6-flow conjecture by proving it for a class of 3-connected graphs. Our method is inspired by Seymour's proof of the 6-flow theorem ( J. Combin. Theory Ser. B 30 (1981), 130–136), and makes use of new connectedness properties of signed graphs.
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