**Conjecture**Every -edge-connected graph can be oriented so that (mod ) for every vertex .

Jaeger called an orientation with the above property a *modular* -*orientation*, and observed that a graph has a modular -orientation if and only if it has a -flow. Thus, this conjecture may be seen as a sharp form of the 2+epsilon flow conjecture. For k=1, this problem is precisely the 3-flow conjecture, and for k=2, Jaeger showed that this conjecture (if true) would imply the 5-flow conjecture. If true, this conjecture would be best possible for every value of k.

The restriction of this conjecture to planar graphs is open, and has a dual formulation. See Mapping planar graphs to odd cycles.

## Bibliography

[J] F. Jaeger, On circular flows in graphs. Finite and infinite sets, Vol. I, II (Eger, 1981), 391--402, Colloq. Math. Soc. János Bolyai, 37, North-Holland, Amsterdam, 1984.. MathSciNet

* indicates original appearance(s) of problem.