Skip to main content
SpringerLink
Log in

Two Equivalent Properties of \(\mathcal{Z}_3\)-Connectivity

  • Original Paper
  • Published:
Graphs and Combinatorics Aims and scope Submit manuscript

Abstract

The concept of group connectivity was introduced by Jaeger et al. (J Comb Theory Ser B 56:165–182, 1992) for the study of integer flows. The concept of all generalized Tutte-orientations was introduced by Barát and Thomassen (J Graph Theory 52:135–146, 2006) for the study of claw-decompositions of graphs. In this paper, we establish the equivalence of the following 3 properties: a graph is \(\mathcal{Z}_3\)-connected, a graph admits all generalized Tutte-orientations and a graph is 3-flow contractible. We also give some applications of this result.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Barát J., Thomassen C.: Claw-decompositions and Tutte-orientations. J. Graph Theory 52, 135–146 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bondy J.A., Murty U.S.R.: Graph Theory With Applications. American Elsevier, New York (1976)

    MATH  Google Scholar 

  3. Catlin P.A.: Double cycle covers and the Petersen graph. J. Graph Theory 13, 465–483 (1989)

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  5. DeVos M., Xu R., Yu G.: Nowhere-zero \(\mathcal{Z}_3\)-flows through \(\mathcal{Z}_3\)-conncetivity. Discrete Math. 306, 26–30 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  6. 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(6), 1325–1336 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  7. 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)

    Article  MathSciNet  MATH  Google Scholar 

  8. Lai H.-J., Xu R., Zhou J.: On group connectivity of graphs. Graphs Comb. 24, 195–203 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  9. Lai H.-J., Yao X.-J.: Group connectivity of graphs with diameter at most 2. Eur. J. Comb. 27, 436–443 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  10. Luo R., Xu R., Yin J.-H., Yu G.: Ore-condition and \(\mathcal{Z}_3\)-connectivity. Eur. J. Comb. 29, 1587–1595 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  11. Luo, R., Xu, R., Yu, G.: An extremal problem on group connectivity of graphs. Eur. J. Comb. (accepted)

  12. Xu R., Zhang C.-Q.: Nowhere-zero 3-flows in squares of graphs. Electron. J. Comb. 10, R5 (2003)

    Google Scholar 

  13. Steinberg R., Younger D.H.: Grötzsch’s theorem for the projective plane. Ars. Comb. 28, 15–31 (1989)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  15. Tutte W.T.: A contributation on the theory of chromatic polynomial. Can. J. Math. 6, 80–91 (1954)

    Article  MathSciNet  MATH  Google Scholar 

  16. Zhang C.-Q.: Integer Flows and Cycle Covers of Graphs. Marcel Dekker, New York (1997)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of West Georgia, Carrollton, GA, 30118, USA

    Rui Xu

Authors
  1. Rui Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Rui Xu.

Additional information

Research is partially supported by the 2011–2012 COSM Research Incentive Award of UWG.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Xu, R. Two Equivalent Properties of \(\mathcal{Z}_3\)-Connectivity. Graphs and Combinatorics 29, 1983–1987 (2013). https://doi.org/10.1007/s00373-012-1197-4

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00373-012-1197-4

Keywords

Search

Navigation