Skip to main content
SpringerLink
Log in

Minimum Degree Conditions for the Proper Connection Number of Graphs

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

Abstract

An edge-coloured graph G is called properly connected if any two vertices are connected by a path whose edges are properly coloured. The proper connection number of a graph G, denoted by pc(G), is the smallest number of colours that are needed in order to make G properly connected. In this paper, we consider sufficient conditions in terms of the ratio between minimum degree and order of a 2-connected graph G implying that G has proper connection number 2.

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

Similar content being viewed by others

References

  1. Andrews, E., Lumduanhom, C., Laforge, E., Zhang, P.: On proper-path colourings in graphs. JCMCC 97, 189–207 (2016)

    MATH  Google Scholar 

  2. Borozan, V., Fujita, S., Gerek, A., Magnant, C., Manoussakis, Y., Montero, L., Tuza, Z.: Proper connection of graphs. Discrete Math. 312(17), 2550–2560 (2012)

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  4. Harary, F.: Graph Theory. Addison-Wesley, Reading, Mass (1969)

  5. Jackson, B.: Long cycles in bipartite graphs. J. Comb. Theory Ser. B 38(2), 118–131 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  6. Menger, K.: Zur allgemeinen Kurventheorie. Fund. Math. 10, 96–115 (1927)

    MATH  Google Scholar 

  7. Paulraja, P.: A characterization of Hamiltonian prisms. J. Graph Theory 17(2), 161–171 (1993)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Discrete Mathematics and Algebra, TU Bergakademie Freiberg, Freiberg, Germany

    Christoph Brause, Trung Duy Doan & Ingo Schiermeyer

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

    Trung Duy Doan

Authors
  1. Christoph Brause
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Trung Duy Doan
    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 Christoph Brause.

Additional information

An extended abstract of this paper has been published in the proceedings of the Cologne Twente Workshop 2016 [3].

Trung Duy Doan: Financial support by the Free State of Saxony (Landesgraduiertenstipendium) in Germany and the National Foundation for Science and Technology Development (NAFOSTED) of Vietnam with project code 101.99-2016.20 is thankfully acknowledged.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Brause, C., Doan, T.D. & Schiermeyer, I. Minimum Degree Conditions for the Proper Connection Number of Graphs. Graphs and Combinatorics 33, 833–843 (2017). https://doi.org/10.1007/s00373-017-1796-1

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00373-017-1796-1

Keywords

Search

Navigation