Tutte's 5-flow conjecture for highly cyclically connected cubic graphs

In 1954, Tutte conjectured that every bridgeless graph has a nowherezero 5-flow. Let ω 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 ω ≥ 2. We show that if a cubic graph G has no edge cut with fewer than 5 2ω − 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 ≥ 52ω − 1, then G has a nowhere-zero 5-flow.