Nowhere-zero Z3Z3-flows through Z3Z3-connectivity

Let ΓΓ be an abelian group. Jaeger et al. [Group connectivity of graphs—a nonhomogeneous analogue of nowhere-zero flow properties, J. Combin. Theory Ser. B 56 (1992) 165–182] introduced a class of graphs which they call ΓΓ-connected. The main interest in ΓΓ-connected graphs is that every ΓΓ-connected graph admits a nowhere-zero ΓΓ-flow. In this paper, we found some families of Z3Z3-connected graphs. Our results generalize an early theorem by Lai (Nowhere-zero 3-flows in locally connected graphs, J. Graph Theory 42 (2003) 211–219) for nowhere-zero Z3Z3-flows in locally connected graphs, and provide a simplified proof of a theorem of Xu and Zhang on nowhere-zero Z3Z3-flows in squares of graphs.