  Tutte's 5-Flow Conjecture from 1954 states that every bridgeless graph has a
nowhere-zero 5-flow. In 2004, Kochol proved that the conjecture is equivalent
to its restriction on cyclically 6-edge connected cubic graphs. We prove that
every cyclically 6-edge-connected cubic graph with oddness at most 4 has a
nowhere-zero 5-flow.
