  In 1954, Tutte conjectured that every bridgeless graph has a nowhere-zero
5-flow. Let $\omega$ be the minimum number of odd cycles in a 2-factor of a
bridgeless cubic graph. Tutte's conjecture is equivalent to its restriction to
cubic graphs with $\omega \geq 2$. We show that if a cubic graph $G$ has no
edge cut with fewer than $ {5/2} \omega - 1$ edges that separates two odd
cycles of a minimum 2-factor of $G$, then $G$ has a nowhere-zero 5-flow. This
implies that if a cubic graph $G$ is cyclically $n$-edge connected and $n \geq
{5/2} \omega - 1$, then $G$ has a nowhere-zero 5-flow.
