Skip to main content
SpringerLink
Log in

Application of soft sets to graph coloring

  • Published:
Computational and Applied Mathematics Aims and scope Submit manuscript

Abstract

Zadeh’s fuzzy set maybe considered as a special case of the soft sets. Soft sets proposed by Molodtsov in 1999 to address uncertainty in a parametric manner. The concept of soft graph was introduced by combining the concepts of graphs and soft sets. In this paper, the notion of graph coloring is extended to soft graphs and then the introduction and investigation of (total) soft coloring and (total) soft chromatic number of soft graphs, which are generalizations of (total) vertex coloring and (total) chromatic number of graphs, respectively, are presented. It is demonstrated that the soft coloring unifies the concepts of vertex coloring, edge coloring, and coloring of hypergraphs. Additionally, the determination of total soft chromatic numbers for certain classes of graphs is provided. In particular, we obtain the lower and upper bounds of set chromatic number of the soft graphs.

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

Similar content being viewed by others

References

  • Akram M, Nawaz S (2015) Operations on soft graphs. Fuzzy Inf Eng 7(4):423–449

    Article  MathSciNet  Google Scholar 

  • Akram M, Nawaz S (2016) Certain types of soft graphs, Politehnica University of Bucharest. Sci Bull Ser A Appl Math Phys 78(4):67–82

    MathSciNet  Google Scholar 

  • Akram M, Zafar F (2015) On soft trees. Bul Acad Stiinte Repub Mold Mat 2(78):82–95

    MathSciNet  Google Scholar 

  • Akram M, Zafar F (2016) Fuzzy soft trees. Southeast Asian Bull Math 40(2):151–170

    MathSciNet  Google Scholar 

  • Al-shami TM, Alcantud JCR, Mhemdi A (2023) New generalization of fuzzy soft sets: \((a; b)\)-Fuzzy soft sets. AIMS Math 8(2):2995–3025

    Article  MathSciNet  Google Scholar 

  • Brooks RL (1941) On colouring the nodes of a network. Math Proc Camb Philos Soc 37(2):194–197

    Article  MathSciNet  Google Scholar 

  • Eslahchi C, Onagh BN (2006) Vertex-strength of fuzzy graphs. Int J Math Math Sci Art ID 43614, 1–9

  • Feng F, Li CX, Davvaz B, Irfan AM (2010) Soft sets combined with fuzzy sets and rough sets: a tentative approach. Soft Comput 14:899–911

    Article  Google Scholar 

  • Feng F, Liu XY, Leoreanu-Fotea V, Jun WB (2011) Soft sets and soft rough sets. Inf Sci 181:1125–1137

    Article  MathSciNet  Google Scholar 

  • Flores-Cruz J, Lara-Velazquez P, Gutierrez-Andrade M, De-los-Cobos-Silva SG, Rincon-Garcia EA (2017) A classifier system using soft graph coloring. Rev Mat Teor Apl 24(1):129–156 (Spanish)

    MathSciNet  Google Scholar 

  • George B, Jose J, Thumbakara RK (2024) Co-normal products and modular products of soft graphs. Discrete Math Algorithms Appl 16(2):Paper No. 2350012, 31

    Article  MathSciNet  Google Scholar 

  • Gong Z, Zhang J (2022) Chromatic number of fuzzy graphs: operations. Fuzzy Graph Color Appl Axioms 11(12):697. https://doi.org/10.3390/axioms11120697

    Article  Google Scholar 

  • Khalaf AB, Ahmed NK, Hamko QH (2022) Soft separation axioms and functions with soft closed graphs. Proyecciones 41(1):177–195

    Article  MathSciNet  Google Scholar 

  • Lara-Velazquez P, Gutierrez-Andrade MA, De-los-Cobos-Silva SG, Rincon-Garcia EA (2015) Soft graph coloring. Rev Mat Teor Apl 22(2):311–323 (Spanish)

    MathSciNet  Google Scholar 

  • Molodtsov D (1999) Soft set theory-first results. Comput Math Appl 37(4–5):19–31

    Article  MathSciNet  Google Scholar 

  • Muoz S, Ortuo MT, Ramrez J, Yez J (2005) Coloring fuzzy graphs. Omega 33:211–221

    Article  Google Scholar 

  • Nawaz HS, Akram M (2021) Oligopolistic competition among the wireless internet service providers of Malaysia using fuzzy soft graphs. J Appl Math Comput 67(1–2):855–890

    Article  MathSciNet  Google Scholar 

  • Rosyida I, Widodo ICR, Indrati RC, Indriati D, Nurhaida N (2020) Fuzzy chromatic number of union of fuzzy graphs: An algorithm, properties and its application. Fuzzy Set Syst 384:2331–2341

    Article  MathSciNet  Google Scholar 

  • Shahzadi S, Sarwar M, Akram M (2020) Decision-making approach with fuzzy type-2 soft graphs. J Math Art. ID 8872446, 25

  • Thumbaraka RK, George B (2014) Soft graphs. Gen Math Notes 21(2):75–86

    Google Scholar 

  • Vizing VG (1965) The chromatic class of a multigraph. Kibernetika 3:29–39

    MathSciNet  Google Scholar 

  • Yolcu A, Benek A, Ozturk TY (2023) A new approach to neutrosophic soft rough sets. Knowl Inf Syst 65:2043–2060

Download references

Acknowledgements

Authors are thankful to the reviewers for their valuable suggestions to improve the paper

Author information

Authors and Affiliations

  1. Department of Mathematics, Payame Noor University, Tehran, Iran

    K. Hamidizadeh, R. Manaviyat & S. Mirvakili

  2. Department of Mathematical Sciences, Yazd University, Yazd, Iran

    B. Davvaz

Authors
  1. K. Hamidizadeh
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. R. Manaviyat
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. S. Mirvakili
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. B. Davvaz
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to B. Davvaz.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest

Human and animal rights

This article does not contain any studies with human participants or animals performed by any of the authors

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 (e.g. a society or other partner) 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

Hamidizadeh, K., Manaviyat, R., Mirvakili, S. et al. Application of soft sets to graph coloring. Comp. Appl. Math. 43, 214 (2024). https://doi.org/10.1007/s40314-024-02738-y

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s40314-024-02738-y

Keywords

Mathematics Subject Classification

Search

Navigation