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

In 1954, Tutte conjectured that every bridgeless graph has a nowhere-zero 5-flow. Let ω ( G ) be the minimum number of odd cycles in a 2-factor of a bridgeless cubic graph G . 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 2 ω ( G ) − 3 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 ≥ 5 2 ω ( G ) − 3 , then G has a nowhere-zero 5-flow.