Nowhere-zero 3-flows and Z3Z3-connectivity in bipartite graphs

Tutte conjectured that every 4-edge-connected graph admits a nowhere-zero 3-flow. Let F12F12 be a family of graphs such that G∈F12G∈F12 if and only if GG is a simple bipartite graph on 12 vertices and δ(G)=4δ(G)=4. Let GG be a simple bipartite graph on nn vertices. It is proved in this paper that if δ(G)≥⌈n4⌉+1, then GG admits a nowhere-zero 3-flow with only one exceptional graph. Moreover, if G∉F12G∉F12 with the minimum degree at least ⌈n4⌉+1 is Z3Z3-connected. The bound is best possible in the sense that the lower bound for the minimum degree cannot be decreased.