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\(Z_3\)-Connectivity of Claw-Free Graphs

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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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References

  1. Bondy, J.A., Murty, U.S.R.: Graph Theory. Springer, Berlin (2008)

    Book  MATH  Google Scholar 

  2. Chen, J., Eschen, E., Lai, H.-J.: Group connectivity of certain graphs. Ars Comb. 89, 141–158 (2008)

    MathSciNet  MATH  Google Scholar 

  3. Devos, M., Xu, R., Yu, G.: Nowhere-zero \(Z_3\)-flows through \(Z_3\)-connectivity. Discret. Math. 306, 26–30 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  4. Fan, G., Zhou, C.: Ore condition and nowhere-zero 3-flows. SIAM J. Discret. Math. 22, 288–294 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  5. Fan, G., Lai, H.-J., Xu, R., Zhang, C.-Q., Zhou, C.: Nowhere-zero 3-flows in triangularly connected graphs. J. Combin. Theory Ser. B 98, 1325–1336 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  6. Jaeger, F., Linial, N., Payan, C., Tarsi, N.: Group connectivity of graphs—a nonhomogeneous analogue of nowhere zero flow properties. J. Combin. Theory Ser. B 56, 165–182 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  7. Kochol, M.: An equivalent version of the 3-flow conjecture. J. Combin. Theory Ser. B 83, 258–261 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  8. Lai, H.-J.: Group connectivity of 3-edge-connected chordal graphs. Graphs Combin. 16, 165–176 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  9. Lai, H.-J.: Nowhere-zero 3-flows in locally connected graphs. J. Graph Theory 42, 211–219 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  10. Lai, H.-J., Li, H., Li, P., Liang, Y., Yao, S.: Mod (2p+1)-orientations in line graphs. Inf. Process. Lett. 111, 1085–1088 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  11. Lai, H.-J., Li, X., Shao, Y., Zhan, M.: Group connectivity and group colorings of graphs—a survey. Acta Math. Sin. (Engl. Ser.) 27, 405–434 (2011)

  12. Li, X., Shao, Y., Lai, H.-J.: Degree condition and \(Z_3\)-connectivity. Discret. Math. 312, 1658–1669 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  13. Lovász, L.M., Thomassen, C., Wu, Y., Zhang, C.-Q.: Nowhere-zero 3-flows and modulo \(k\)-orientations. J. Combin. Theory Ser. B 103, 587–598 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  14. Luo, R., Xu, R., Yin, J., Yu, G.: Ore-condition and \(Z_3\)-connectivity. Eur. J. Combin. 29, 1587–1595 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  15. Ma, J., Li, X.: Nowhere-zero \(3\)-flows of claw-free graphs. Discret. Math. 336, 57–68 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  16. Shi, R.: 2-neighborhoods and Hamiltonian conditions. J. Graph Theory 16, 267–271 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  17. Thomassen, C.: The weak 3-flow conjecture and the weak circular flow conjecture. J. Combin. Theory Ser. B 102, 521–529 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  18. Tutte, W.T.: On the imbedding of linear graphs in surfaces. Proc. Lond. Math. Soc. Ser. 2(51), 474–483 (1949)

    MathSciNet  MATH  Google Scholar 

  19. Tutte, W.T.: A contribution to the theory of chromatic polynomials. Can. J. Math. 6, 80–91 (1954)

    Article  MathSciNet  MATH  Google Scholar 

  20. Yang, F., Li, X.: Degree sum condition for 3-independent set and \(Z_3\)-connectivity. Discret. Math. 313, 2493–2505 (2013)

    Article  MATH  Google Scholar 

  21. Zhang, X., Zhan, M., Xu, R., Shao, Y., Li, X., Lai, H.-J.: Degree Sum condition for \(Z_3\)-connectivity in graphs. Discret. Math. 310, 3390–3397 (2010)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Central China Normal University, Wuhan, 430079, China

    Ziwen Huang, Xiangwen Li & Jianqing Ma

Authors
  1. Ziwen Huang
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  2. Xiangwen Li
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  3. Jianqing Ma
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Corresponding author

Correspondence to Xiangwen Li.

Additional information

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

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