Z3Z3-connectivity of 4-edge-connected 2-triangular graphs

A graph GG is kk-triangular if each edge of GG is in at least kk triangles. It is conjectured that every 44-edge-connected 1-triangular graph admits a nowhere-zero Z3Z3-flow. However, it has been proved that not all such graphs are Z3Z3-connected. In this paper, we show that every 44-edge-connected 2-triangular graph is Z3Z3-connected. The result is best possible. This result provides evidence to support the Z3Z3-connectivity conjecture by Jaeger et al that every 5-edge-connected graph is Z3Z3-connected.