  There are many major open problems in integer flow theory, such as Tutte's
3-flow conjecture that every 4-edge-connected graph admits a nowhere-zero
3-flow, Jaeger et al.'s conjecture that every 5-edge-connected graph is
$Z_3$-connected and Kochol's conjecture that every bridgeless graph with at
most three 3-edge-cuts admits a nowhere-zero 3-flow (an equivalent version of
3-flow conjecture). Thomassen proved that every 8-edge-connected graph is
$Z_3$-connected and therefore admits a nowhere-zero 3-flow. Furthermore,
Lov$\acute{a}$sz, Thomassen, Wu and Zhang improved Thomassen's result to
6-edge-connected graphs. In this paper, we prove that: (1) Every
4-edge-connected graph with at most seven 5-edge-cuts admits a nowhere-zero
3-flow. (2) Every bridgeless graph containing no 5-edge-cuts but at most three
3-edge-cuts admits a nowhere-zero 3-flow. (3) Every 5-edge-connected graph with
at most five 5-edge-cuts is $Z_3$-connected. Our main theorems are partial
results to Tutte's 3-flow conjecture, Kochol's conjecture and Jaeger et al.'s
conjecture, respectively.
