  The study of nowhere-zero flows began with a key observation of Tutte that in
planar graphs, nowhere-zero k-flows are dual to k-colourings (in the form of
k-tensions). Tutte conjectured that every graph without a cut-edge has a
nowhere-zero 5-flow. Seymour proved that every such graph has a nowhere-zero
6-flow.
  For a graph embedded in an orientable surface of higher genus, flows are not
dual to colourings, but to local-tensions. By Seymour's theorem, every graph on
an orientable surface without the obvious obstruction has a nowhere-zero
6-local-tension. Bouchet conjectured that the same should hold true on
non-orientable surfaces. Equivalently, Bouchet conjectured that every
bidirected graph with a nowhere-zero $\mathbb{Z}$-flow has a nowhere-zero
6-flow. Our main result establishes that every such graph has a nowhere-zero
12-flow.
