Two Equivalent Properties of mathcal{Z}_3-Connectivity

The concept of group connectivity was introduced by Jaeger et al. (J Comb Theory Ser B 56:165–182, 1992) for the study of integer flows. The concept of all generalized Tutte-orientations was introduced by Barát and Thomassen (J Graph Theory 52:135–146, 2006) for the study of claw-decompositions of graphs. In this paper, we establish the equivalence of the following 3 properties: a graph is $mathcal{Z}_3$ -connected, a graph admits all generalized Tutte-orientations and a graph is 3-flow contractible. We also give some applications of this result.