Abstract
Let G be a 2-edge-connected simple graph, and let A denote an abelian group with the identity element 0. If a graph G * is obtained by repeatedly contracting nontrivial A-connected subgraphs of G until no such a subgraph left, we say G can be A-reduced to G*. A graph G is bridged if every cycle of length at least 4 has two vertices x, y such that d G (x, y) < d C (x, y). In this paper, we investigate the group connectivity number Λ g (G) = min{n: G is A-connected for every abelian group with |A| ≥ n} for bridged graphs. Our results extend the early theorems for chordal graphs by Lai (Graphs Comb 16:165–176, 2000) and Chen et al. (Ars Comb 88:217–227, 2008).
Similar content being viewed by others
References
Bondy J.A., Murty U.S.R.: Graphs Theroy. Springer, New York (2008)
Chen J., Eschen E., Lai H.-J.: Group connectivity of certain graphs. Ars Comb. 89, 217–227 (2008)
Devos M., Xu R., Yu X.: 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., Linial N., Payan C., Tarsi N.: Group connectivity of graphs-a nonhomogeneous analogue of nowhere zero flow properties. J. Comb. Theory Ser. B 56, 165–182 (1992)
Jensen T.R., Toft B.: Graph Coloring Problems. Wiley, New York (1995)
Lai H.-J., Li X.: Group chromatic number of planar graphs of girth at least 4. J. Graph Theory 52, 51–72 (2003)
Lai H.-J.: Group connectivity of 3-edge-connected chordal graphs. Graphs Comb. 16, 165–176 (2000)
Lai H.-J.: Nowhere-zero 3-flows in locally connected graphs. J. Graph Theory 42, 211–219 (2003)
Luo R., Xu R., Yin J., Yu G.: Ore condition and Z 3-connectivity. Eur. J. Comb. 29, 1587–1595 (2008)
Thomassen C.: Grötzsh’ 3-color theorem and its counterparts for the torus and the projective plane. J. Comb. Theory Ser. B 62, 268–297 (1994)
Tutte W.T.: A contribution on the theory of chromatic polynomial. Can. J. Math 6, 80–91 (1954)
Tutte W.T.: On the algebraic theory of graph colorings. J. Comb. Theory 1, 15–50 (1966)
Yao X., Li X., Lai H.-J: Degree conditions for group connectivity. Discret. Math. 310, 1050–1058 (2010)
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)
Author information
Authors and Affiliations
Corresponding author
Additional information
X. Li was supported by the Natural Science Foundation of China (11171129).
Rights and permissions
About this article
Cite this article
Li, L., Li, X. & Shu, C. Group Connectivity of Bridged Graphs. Graphs and Combinatorics 29, 1059–1066 (2013). https://doi.org/10.1007/s00373-012-1154-2
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00373-012-1154-2