Soft Computing

, Volume 23, Issue 4, pp 1337–1346

# Uncertain vertex coloring problem

• Lin Chen
• Jin Peng
• Dan A. Ralescu
Methodologies and Application

## Abstract

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.

## Keywords

Vertex coloring problem Uncertain graph Maximal uncertain independent vertex set Degree of belief rule Uncertain chromatic set Maximum separation degree algorithm

## Notes

### Acknowledgements

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.

### Conflict of interest

The authors declare that there is no conflict of interests regarding the publication of this paper.

### Ethical approval

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

### Informed consent

Informed consent was obtained from all individual participants included in the study.

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