Skip to main content
SpringerLink
Log in

The Conflict-Free Vertex-Connection Number and Degree Conditions of Graphs

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

Abstract

A path in a vertex-coloured graph is called conflict-free if there is a colour used on exactly one of its vertices. A vertex-coloured graph is said to be conflict-free vertex-connected if any two distinct vertices of the graph are connected by at least one conflict-free vertex-path. The conflict-free vertex-connection number, denoted by vcfc(G), is the smallest number of colours needed in order to make G conflict-free vertex-connected. Our main result of this paper is the following: Let G be a connected graph of order \(n\ge 11\). If \(\delta (G)\ge \frac{n}{5}\), then \(vcfc(G)\le 3\). Moreover, we show that both bounds, on the order n and the minimum degree \(\delta (G)\), are tight. Furthermore, we generalize our result by requiring minimum degree-sum conditions for pairs, triples or quadruples of independent vertices.

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
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11

Similar content being viewed by others

Availability of data and materials

Not applicable.

Code availability

Not applicable.

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, 833–843 (2017). https://doi.org/10.1007/s00373-017-1796-1

    Article  MathSciNet  MATH  Google Scholar 

  3. Brause, C., Doan, T.D., Schiermeyer, I.: On the minimum degree and the proper connection number of graphs. Electron. Notes Discrete Math. 55, 109–112 (2016)

    Article  MATH  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  MATH  Google Scholar 

  5. Chartrand, G., Johns, G.L., McKeon, K.A., Zhang, P.: Rainbow connection in graphs. Math. Bohemica 133(1), 85–98 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  6. Chizmar, E., Magnant, C., Nowbandegani, P.S.: Note on vertex and total proper connection numbers. AKCE Int. J. Graphs Combin. 13(2), 103–106 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chang, H., Doan, T.D., Huang, Z., Jendrol’, S., Li, X., Schiermeyer, I.: Graphs with conflict-free connection number two. Graphs Combin. 34, 1553–1563 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  8. Czap, J., Jendrol’, S., Valiska, J.: Conflict-free connections of graphs. Discuss. Math. Graph Theory 38(4), 911–920 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  9. Chen, L., Li, X., Shi, Y.: The complexity of determining the rainbow vertex-connection of a graph. Theor. Comput. Sci. 412(35), 4531–4535 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  10. Doan, T.D., Schiermeyer, I.: Conflict-free vertex connection number at most \(3\) and size of graphs. Discuss. Math. Graph Theory 41, 617–632 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  11. Doan, T.D., Schiermeyer, I.: Conflict-free connection number and size of graphs. Graphs Combin. 37, 1859–1871 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  12. Jiang, H., Li, X., Zhang, Y., Zhao, Y.: On (Strong) proper vertex-connection of graphs. Bull. Malays. Math. Sci. Soc. 41(1), 415–425 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  13. Krivelevich, M., Yuster, R.: The rainbow connection of a graph is (at most) reciprocal to its minimum degree. J. Graph Theory 63(3), 185–191 (2010)

    MathSciNet  MATH  Google Scholar 

  14. Li, X., Shi, Y., Sun, Y.: Rainbow connections of graphs: a survey. Graphs Combin. 29, 1–38 (2013)

    Article  MathSciNet  MATH  Google Scholar 

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

    Book  Google Scholar 

  16. Li, X., Magnant, C.: Properly colored notions of connectivity—a dynamic survey. Theory Appl. Graphs 0(1) (2015) (Art. 2)

  17. Li, X., Zhang, Y., Zhu, X., Mao, Y., Zhao, H., Jendrol’, S.: Conflict-free vertex-connections of graphs. Discuss. Math. Graph Theory 40, 51–65 (2020)

  18. Li, Z., Wu, B.: On the maximum value of conflict-free vertex-connection number of graphs. Discrete Math. Algor. Appl. 10(05) (2018)

  19. West, D.B.: Introduction to Graph Theory. Prentice Hall, Hoboken (2001)

    Google Scholar 

Download references

Acknowledgements

We would like to thank the two reviewers for several valuable comments and suggestions improving this paper. A part of work was completed during a stay of the second author at the Vietnam Institute for Advanced Study in Mathematics (VIASM). He would like to thank this institution for financial support and hospitality.

Funding

This research was partly supported by Vietnam Ministry of Education and Training under grant number B2022-BKA-03.

Author information

Authors and Affiliations

  1. School of Applied Mathematics and Informatics, Hanoi University of Science and Technology, Hanoi, Vietnam

    Trung Duy Doan

  2. Department of Mathematics, Hanoi National University of Education, 136 XuanThuy Street, Hanoi, Vietnam

    Pham Hoang Ha

  3. Institut für Diskrete Mathematik und Algebra, Technische Universität Bergakademie Freiberg, 09596, Freiberg, Germany

    Ingo Schiermeyer

Authors
  1. Trung Duy Doan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Pham Hoang Ha
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ingo Schiermeyer
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Trung Duy Doan.

Ethics declarations

Conflict of interest

Not applicable.

Additional information

Publisher's Note

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

Rights and permissions

Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Doan, T.D., Ha, P.H. & Schiermeyer, I. The Conflict-Free Vertex-Connection Number and Degree Conditions of Graphs. Graphs and Combinatorics 38, 157 (2022). https://doi.org/10.1007/s00373-022-02567-y

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00373-022-02567-y

Keywords

Mathematics Subject Classification 2010

Search

Navigation