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Jaeger et al. conjectured that every 5-edge-connected graph is $$Z_3$$ Z 3 -connected, which is equivalent to that every 5-edge-connected claw-free graph is $$Z_3$$ Z 3 -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...
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