Nowhere-zero 5-flows on cubic graphs with oddness 4
  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. 
Giuseppe Mazzuoccolo and Eckhard Steffen
math.CO