Jaeger et al. conjectured that every 5-edge-connected graph is $$Z_3$$Z3-connected, which is equivalent to that every 5-edge-connected claw-free graph is $$Z_3$$Z3-connected by Lai et al. (Inf Process Lett 111:1085---1088, 2011), and Ma and Li (Discret Math 336:57---68, 2014). Let G be a claw-free graph on at least 3 vertices such that there are at least two common neighbors of every pair of 2-distant vertices. In this paper, we prove that G is not $$Z_3$$Z3-connected if and only if G is one of seven specified graphs, or three families of well characterized graphs. As a corollary, G does not a...
