Abstract
Jaeger et al. (J Comb Theory Ser B 56:165–182, 1992) conjectured that every 5-edge-connected graph is \(Z_3\)-connected. Moreover, Lai et al. (Discret Math 311:2295–2307, 2011) proved that every 5-edge-connected graph is \(Z_3\)-connected if and only if every 5-edge-connected line graph is \(Z_3\)-connected. A graph \(G\) is a \(J_3\) graph if every edge of \(G\) lies in a 3-cycle of \(G\). We prove that every 5-edge-connected \(J_3\) line graph is \(Z_3\)-connected.
Similar content being viewed by others
References
Bondy, J.A., Murty, U.S.R.: Graph Theory with Applications. American Elsevier, New York (1976)
Chen, J.J., Eschen, E., Lai, H.-J.: Group connectivity of certain graphs. Ars Comb. 89, 141–158 (2008)
Chen, Z.-H., Lai, H.-J., Lai, H.Y.: Nowhere zero flows in line graphs. Discret. Math. 230, 133–141 (2001)
DeVos, M., Xu, R., Yu, G.: Nowhere-zero \(Z_3\)-flows through \(Z_3\)-connectivity. Discret. Math. 306, 26–30 (2006)
Fan, G., Lai, H.-J., Xu, R., Zhang, C.-Q., Zhou, C.: Nowhere-zero 3-flows in triangularly connected graphs. J. Comb. Theory Ser. B 98, 1325–1336 (2008)
Jaeger, F.: Nowhere-zero flow problems. In: Beineke, L., Wilson, R. (eds.) Selected Topics in Graph Theory, vol. 3, pp. 91–95. Academic Press, London (1988)
Jaeger, F., Linial, N., Payan, C., Tarsi, M.: Group connectivity of graphs—a nonhomogeneous analogue of nowhere-zero flow properties. J. Comb. Theory Ser. B 56,165–182 (1992)
Kochol, M.: An equivalent version of the 3-flow conjecture. J. Comb. Theory Ser. B 83, 258–261 (2001)
Lai, H.-J.: Group connectivity of 3-edge-connected chordal graphs. Graphs Comb. 16, 165–176 (2000)
Lai, H.-J., Li, H., Li, P., Liang, Y., Yao, S.: Group connectivity in line graphs. Discret. Math. 311, 2295–2307 (2011)
Lai, H.-J., Miao, L., Shao, Y.: Every line graph of a 4-edge-connected graph is \(Z_3\)-connected. Eur. J. Comb. 30, 595–601 (2009)
Lai, H.-J., Xu, R., Zhou, J.: On Group connectivity of graphs. Graphs Comb. 24, 1–9 (2008)
Lovász, L.M., Thomassen, C., Wu, Y., Zhang, C.-Q.: Nowhere-zero 3-flows and modulo \(k\)-orientations. J. Comb. Theory Ser. B 103, 587–598 (2013)
Thomassen, C.: The weak 3-flow conjecture and the weak circular flow conjecture. J. Comb. Theory B 102, 521–529 (2012)
Tutte, W.T.: A contribution to the theory of chromatic polynomials. Can. J. Math. 6, 80–91 (1954)
Tutte, W.T.: On the algebraic theory of graph colourings. J. Comb. Theory 1, 15–50 (1996)
Xu, R., Zhang, C.-Q.: Nowhere-zero 3-flows in squares of graphs. Electron. J. Comb. 10, R5 (2003)
Acknowledgments
The authors would like to thank Professor Lai for the valuable comments. The first author was supported by National Science Foundation of China (11326215). The second author was supported by National Science Foundation of China (11171129).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Yang, F., Li, X. Group connectivity in \(J_3\) line graphs. Graphs and Combinatorics 31, 1065–1076 (2015). https://doi.org/10.1007/s00373-014-1418-0
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00373-014-1418-0