Abstract
Jaeger et al. conjectured that every 5-edge-connected graph is \(Z_3\)-connected, which is equivalent to that every 5-edge-connected claw-free graph is \(Z_3\)-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 of every pair of 2-distant vertices. In this paper, we prove that G is not \(Z_3\)-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 admit a nowhere-zero 3-flow if and only if G is one of three specified graphs or a family of well characterized graphs.
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X. Li was supported by the Natural Science Foundation of China (11571134) and by Doctoral Fund of Ministry of Education of China (20130144110001).
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Huang, Z., Li, X. & Ma, J. \(Z_3\)-Connectivity of Claw-Free Graphs. Graphs and Combinatorics 33, 123–140 (2017). https://doi.org/10.1007/s00373-016-1754-3
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DOI: https://doi.org/10.1007/s00373-016-1754-3