Uncertain vertex coloring problem
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This paper investigates the vertex coloring problem in an uncertain graph in which all vertices are deterministic, while all edges are not deterministic and exist with some degree of belief in uncertain measures. The concept of the maximal uncertain independent vertex set of an uncertain graph is first introduced. We then present a degree of belief rule to obtain the family of maximal uncertain independent vertex sets. Based on the maximal uncertain independent vertex set, some properties of the separation degree of an uncertain graph are discussed. Following that, the concept of an uncertain chromatic set is introduced. Then, a maximum separation degree algorithm is derived to obtain the uncertain chromatic set. Finally, numerical examples are presented to demonstrate the application of the vertex coloring problem in uncertain graphs and the effectiveness of the maximum separation degree algorithm.
KeywordsVertex coloring problem Uncertain graph Maximal uncertain independent vertex set Degree of belief rule Uncertain chromatic set Maximum separation degree algorithm
This work was supported by the National Natural Science Foundation of China (Nos. 11626234, 61703438), and the Key Project of Hubei Provincial Natural Science Foundation (No. 2015CFA144), China.
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Conflict of interest
The authors declare that there is no conflict of interests regarding the publication of this paper.
This article does not contain any studies with human participants or animals performed by any of the authors.
Informed consent was obtained from all individual participants included in the study.
- Chow F, Hennessy J (1990) The priority-based coloring approach to register allocation. ACM Trans Program Lang Syst 12(4):501–536Google Scholar
- Erdős P, Rényi A (1959) On random graph. Publ Math 6:290–297Google Scholar
- Floyd R (1962) Algorithm 97: shortest path. Commun ACM 5(6):345Google Scholar
- Gao Y, Qin Z (2016) On computing the edge-connectivity of an uncertain graph. IEEE Trans Fuzzy Syst 24(4):981–991Google Scholar
- Liu B (2009) Some research problems in uncertainty theory. J Uncertain Syst 3(1):3–10Google Scholar
- Liu B (2012) Why is there a need for uncertainty theory. J Uncertain Syst 6(1):3–10Google Scholar
- Muñoz S, Ortuño M, Ramírez J, Yáñez J (2005) Coloring fuzzy graphs. Omega 33(3):211–221Google Scholar
- Rosenfeld A (1975) Fuzzy graphs. In: Zadeh L, Fu K, Shimura M (eds) Fuzzy sets and their applications to cognitive and decision processes. Academic Press, New York, pp 77–95Google Scholar
- Rosyida I, Widodo, Indrati C, Sugeng K (2016a) An \(\alpha \)-cut chromatic number of a total uncertain graph and its properties. In: Proceedings of the 7th SEAMS UGM international conference on mathematics and its applications 2015, pp 1–8Google Scholar
- Rosyida I, Peng J, Chen L, Widodo, Indrati C, Sugeng K (2016b) An uncertain chromatic number of an uncertain graph based on \(\alpha \)-cut coloring. Fuzzy Optim Decis Making. doi: 10.1007/s10700-016-9260-x