  Tutte conjectured in 1972 that every 4-edge connected graph has a
nowhere-zero 3-flow. This has long been known to be equivalent to the
conjecture that every 5-regular 4-edge-connected graph has an edge orientation
in which every out-degree is either 1 or 4. We show that the assertion of the
conjecture holds asymptotically almost surely for random 5-regular graphs. It
follows that the conjecture holds for almost all 4-edge connected 5-regular
graphs.
