Skip to main content
SpringerLink
Log in

Two Sufficient Conditions for 2-Connected Graphs to Have Proper Connection Number 2

  • Published:
Bulletin of the Malaysian Mathematical Sciences Society Aims and scope Submit manuscript

Abstract

The proper connection number of a graph G, denoted by pc(G), is the minimum number of colors needed to color the edges of G so that every pair of distinct vertices of G is connected by a path in which no two adjacent edges of the path receive the same color. A connected graph G is k-PC if pc(G)  ≤  k. In the literature, it is shown that every 3-edge-connected graph is 2-PC and every 2-edge-connected graph is 3-PC. We show in this paper that every 2-connected graph such that each edge is contained in a cycle of length at most 4 is 2-PC, and every 2-connected graph with minimum degree at least 3 such that each edge is contained in a cycle of length at most 6 is 2-PC.

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

Fig. 1
Fig. 2

Similar content being viewed by others

References

  1. Andrews, E., Laforge, E., Lumduanhom, C., Zhang, P.: On proper-path colorings in graphs. J. Combin. Math. Combin. Comput. 97, 189–207 (2016)

    MathSciNet  MATH  Google Scholar 

  2. Brause, C., Doan, T.D., Schiermeyer, I.: Minimum degree conditions for the proper connection number of graphs. Graphs Combin. 33(4), 833–843 (2017)

    Article  MathSciNet  Google Scholar 

  3. Bondy, J.A., Murty, U.S.R.: Graph Theory, Graduate Texts in Mathematics, vol. 244. Springer, London (2008)

    Google Scholar 

  4. Borozan, V., Fujita, S., Gerek, A., Magnant, C., Manoussakis, Y., Montero, L., Tuza, Z.: Proper connection of graphs. Discrete Math. 312, 2550–2560 (2012)

    Article  MathSciNet  Google Scholar 

  5. Chartrand, G., Johns, G.L., McKeon, K.A., Zhang, P.: Rainbow connection in graphs. Math. Bohem. 133, 85–98 (2008)

    MathSciNet  MATH  Google Scholar 

  6. Chou, W.S., Manoussakis, Y., Megalakaki, O., Spyratos, M., Tuza, Z.: Paths through fixed vertices in edge-colored graphs. Math. Sci. Hum. 127, 49–58 (1994)

    MathSciNet  MATH  Google Scholar 

  7. Dorninger, D.: On Permutations of Chromosomes, Contributions to General Algebra 5 (Salzburg, 1986), pp. 95–103. Hölder-Pichler-Tempsky, Vienna (1987)

    Google Scholar 

  8. Dorninger, D.: Hamiltonian circuits determining the order of chromosomes. Discrete Appl. Math. 50, 159–168 (1994)

    Article  MathSciNet  Google Scholar 

  9. Dorninger, D., Timischl, W.: Geometrical constraints on Bennett’s predictions of chromosome order. Heredity 58, 321–325 (1987)

    Article  Google Scholar 

  10. Gu, R., Li, X.L., Qin, Z.M.: Proper connection number of random graphs. Theor. Comput. Sci. 609, 336–343 (2016)

    Article  MathSciNet  Google Scholar 

  11. Huang, F., Li, X.L., Qin, Z.M., Magnant, C.: On two conjectures about the proper connection number of graphs. Discrete Math. 340, 2217–2222 (2017)

    Article  MathSciNet  Google Scholar 

  12. Huang, F., Li, X.L., Wang, S.J.: Upper bound of proper connection number of graphs. J. Comb. Optim. 34(1), 165–173 (2017)

    Article  MathSciNet  Google Scholar 

  13. Huang, F., Yuan, J.J.: On strong proper connection number of cubic graphs. Discrete Appl. Math. 265, 104–119 (2019)

    Article  MathSciNet  Google Scholar 

  14. Li, X.L., Magnant, C.: Properly colored notions of connectivity—a dynamic survey. Theory Appl. Graphs 0(1), 2 (2015). https://doi.org/10.20429/tag.2015.000102

    Article  Google Scholar 

  15. Li, X.L., Wei, M.Q., Yue, J.: Proper connection number and connected dominating sets. Theor. Comput. Sci. 607, 480–487 (2015)

    Article  MathSciNet  Google Scholar 

  16. Li, X.L., Magnant, C., Qin, Z.M.: Properly Colored Connectivity of Graphs, Springer Briefs in Mathematics. Springer, Cham (2018)

    Book  Google Scholar 

  17. Li, X.L., Shi, Y.T., Sun, Y.F.: Rainbow connections of graphs: a survey. Graphs Comb. 29, 1–38 (2013)

    Article  MathSciNet  Google Scholar 

  18. Li, X.L., Sun, Y.T.: Rainbow Connections of Graphs, Springer Briefs in Mathematics. Springer, New York (2012)

    Book  Google Scholar 

  19. Westbrook, J., Tarjan, R.E.: Maintaining bridge-connected and biconnected components on-line. Algorithmica 7(1–6), 433–464 (1992)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the associate editor and two anonymous referees for their constructive comments and helpful suggestions. This research was supported by NSFC under Grant Numbers 11801526 and 11671368.

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Zhengzhou University, Zhengzhou, 450001, Henan, People’s Republic of China

    Shanshan Guo, Fei Huang & Jinjiang Yuan

Authors
  1. Shanshan Guo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Fei Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jinjiang Yuan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Fei Huang.

Additional information

Communicated by Sanming Zhou.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Guo, S., Huang, F. & Yuan, J. Two Sufficient Conditions for 2-Connected Graphs to Have Proper Connection Number 2. Bull. Malays. Math. Sci. Soc. 43, 3323–3331 (2020). https://doi.org/10.1007/s40840-019-00868-9

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40840-019-00868-9

Keywords

Mathematics Subject Classification

Search

Navigation