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.
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References
Bondy, J., & Murty, U. (1976). Graph theory with applications. New York: Elsevier.
Brown, J. (1972). Chromatic scheduling and the chromatic number problem. Management Science, 19(4), 456–563.
Chen, L., & Peng, J. (2014). Vertex coloring for uncertain graph. In Proceedings of the sixteenth national youth conference on information and management sciences (pp. 191–199). Luoyang, China.
Chen, L., Peng, J., Zhang, B., & Li, S. (2014). Uncertain programming model for uncertain minimum weight vertex covering problem. Journal of Intelligent Manufacturing. doi:10.1007/s10845-014-1009-1.
Erdős, P., & Rényi, A. (1959). On random graphs. Publicationes Mathematicae, 6, 290–297.
Gao, X. (2014). Regularity index of uncertain graph. Journal of Intelligent and Fuzzy Systems, 27(4), 1671–1678.
Gao, X. (2016). Tree index of uncertain graphs. Soft Computing, 20(4), 1449–1458.
Gao, X., & Gao, Y. (2013). Connectedness index of uncertain graph. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 21(1), 127–137.
Gao, Y., Yang, L., Li, S., & Kar, S. (2015). On distribution function of the diameter in uncertain graph. Information Sciences, 296, 61–74.
Gilbert, E. (1959). Random graphs. Annals of Mathematical Statistics, 30(4), 1141–1144.
Ji, X., & Zhou, J. (2015). Multi-dimensional uncertain differential equation: Existence and uniqueness of solution. Fuzzy Optimization and Decision Making, 14(4), 477–491.
Liu, B. (2007). Uncertainty theory (2nd ed.). Berlin: Springer.
Liu, B. (2009). Some research problems in uncertainty theory. Journal of Uncertain Systems, 3(1), 3–10.
Liu, B. (2010). Uncertainty theory: A branch of mathematics for modeling human uncertainty. Berlin: Springer.
Liu, B. (2014). Uncertainty distribution and independence of uncertain processes. Fuzzy Optimization and Decision Making, 13(3), 259–271.
Muñoz, S., Ortuño, M., Ramirez, J., & Yañez, J. (2005). Coloring fuzzy graphs. Omega, 33(3), 211–221.
Sommer, C. (2009). A note on coloring sparse random graphs. Discrete Mathematics, 39(2), 3381–3384.
Wu, X., Zhao, R., & Tang, W. (2014). Uncertain agency models with multi-dimensional incomplete information based on confidence level. Fuzzy Optimization and Decision Making, 13(2), 231–258.
Yang, K., Zhao, R., & Lan, Y. (2016). Incentive contract design in project management with serial tasks and uncertain completion times. Engineering Optimization, 48(4), 629–651.
Yao, K. (2015). A no-arbitrage theorem for uncertain stock model. Fuzzy Optimization and Decision Making, 14(2), 227–242.
Yao, K. (2015). Uncertain contour process and its application in stock model with floating interest rate. Fuzzy Optimization and Decision Making, 14(4), 399–424.
Zadeh, L. (1965). Fuzzy sets. Information and Control, 8(3), 338–353.
Zhang, B., & Peng, J. (2012). Euler index in uncertain graph. Applied Mathematics and Computation, 218(20), 10279–10288.
Zhang, B., & Peng, J. (2013). Connectedness strength of two vertices in an uncertain graph. International Journal of Computer Mathematics, 90(2), 246–257.
Zhang, B., & Peng, J. (2013). Matching index of uncertain graph: Concept and algorithm. Applied and Computational Mathematics, 12(3), 381–391.
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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Rosyida, I., Peng, J., Chen, L. et al. An uncertain chromatic number of an uncertain graph based on \(\alpha \)-cut coloring . Fuzzy Optim Decis Making 17, 103–123 (2018). https://doi.org/10.1007/s10700-016-9260-x
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DOI: https://doi.org/10.1007/s10700-016-9260-x