Fuzzy Optimization and Decision Making

, Volume 17, Issue 1, pp 103–123 | Cite as

An uncertain chromatic number of an uncertain graph based on \(\alpha \)-cut coloring

  • Isnaini Rosyida
  • Jin Peng
  • Lin Chen
  • Widodo Widodo
  • Ch. Rini Indrati
  • Kiki A. Sugeng
Article

Abstract

An uncertain graph is a graph in which the edges are indeterminate and the existence of edges are characterized by belief degrees which are uncertain measures. This paper aims to bring graph coloring and uncertainty theory together. A new approach for uncertain graph coloring based on an \(\alpha \)-cut of an uncertain graph is introduced in this paper. Firstly, the concept of \(\alpha \)-cut of uncertain graph is given and some of its properties are explored. By means of \(\alpha \)-cut coloring, we get an \(\alpha \)-cut chromatic number and examine some of its properties as well. Then, a fact that every \(\alpha \)-cut chromatic number may be a chromatic number of an uncertain graph is obtained, and the concept of uncertain chromatic set is introduced. In addition, an uncertain chromatic algorithm is constructed. Finally, a real-life decision making problem is given to illustrate the application of the uncertain chromatic set and the effectiveness of the uncertain chromatic algorithm.

Keywords

Uncertain graph \(\alpha \)-cut of uncertain graph \(\alpha \)-cut coloring \(\alpha \)-cut chromatic number Uncertain chromatic set 

Notes

Acknowledgements

This work was supported by the Sandwich-Like scholarship (No. 1181/E4.4/K/2014) from the Directorate General of Higher Education, Ministry of Education and Culture of Indonesia. This work was also supported by the Projects of the Humanity and Social Science Foundation of Ministry of Education of China (No. 13YJA630065), and the Key Project of Hubei Provincial Natural Science Foundation (No. 2015CFA144), China.

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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Isnaini Rosyida
    • 1
    • 2
  • Jin Peng
    • 3
  • Lin Chen
    • 4
  • Widodo Widodo
    • 1
  • Ch. Rini Indrati
    • 1
  • Kiki A. Sugeng
    • 5
  1. 1.Department of MathematicsGadjah Mada UniversityYogyakartaIndonesia
  2. 2.Department of MathematicsSemarang State UniversitySemarangIndonesia
  3. 3.Institute of Uncertain SystemsHuanggang Normal UniversityHubeiChina
  4. 4.College of Management and EconomicsTianjin UniversityTianjinChina
  5. 5.Department of MathematicsUniversity of IndonesiaDepokIndonesia

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