Skip to main content
SpringerLink
Log in

Uncertain vertex coloring problem

  • Methodologies and Application
  • Published:
Soft Computing Aims and scope Submit manuscript

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.

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.

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

References

  • Appel K, Haken W (1976) Every planar map is four colorable. Bull Am Math Soc 82(5):711–712

    Article  MathSciNet  MATH  Google Scholar 

  • Bollobás B (1988) The chromatic number of random graphs. Combinatorica 8(1):49–55

    Article  MathSciNet  MATH  Google Scholar 

  • Carrabs F, Cerulli R, Gentili M (2009) The labeled maximum matching problem. Comput Oper Res 36(6):1859–1871

    Article  MathSciNet  MATH  Google Scholar 

  • Cerulli R, Donato R, Raiconi A (2012) Exact and heuristic methods to maximize network lifetime in wireless sensor networks with adjustable sensing ranges. Eur J Oper Res 220(1):58–66

    Article  MathSciNet  MATH  Google Scholar 

  • Chow F, Hennessy J (1990) The priority-based coloring approach to register allocation. ACM Trans Program Lang Syst 12(4):501–536

    Article  Google Scholar 

  • Demange M, Ekim T, Ries B (2016) On the minimum and maximum selective graph coloring problems in some graph classes. Discret Appl Math 204:77–89

    Article  MathSciNet  MATH  Google Scholar 

  • Eisenbrand F, Niemeier M (2012) Coloring fuzzy circular interval graphs. Eur J Comb 33(5):893–904

    Article  MathSciNet  MATH  Google Scholar 

  • Erdős P, Rényi A (1959) On random graph. Publ Math 6:290–297

    Google Scholar 

  • Floyd R (1962) Algorithm 97: shortest path. Commun ACM 5(6):345

    Article  Google Scholar 

  • Furini F, Gabrel V, Ternier I (2017) An improved DSATUR-based branch-and-bound algorithm for the vertex coloring problem. Networks 69(1):124–141

    Article  MathSciNet  MATH  Google Scholar 

  • Gao X (2016) Tree index of uncertain graphs. Soft Comput 20(4):1449–1458

    Article  MATH  Google Scholar 

  • Gao X, Gao Y (2013) Connectedness index of uncertain graph. Int J Uncertain Fuzziness Knowl Based Syst 21(1):127–137

    Article  MathSciNet  MATH  Google Scholar 

  • Gao Y, Qin Z (2016) On computing the edge-connectivity of an uncertain graph. IEEE Trans Fuzzy Syst 24(4):981–991

    Article  Google Scholar 

  • Gao Y, Yang L, Li S, Kar S (2015) On distribution function of the diameter in uncertain graph. Inf Sci 296:61–74

    Article  MathSciNet  MATH  Google Scholar 

  • Gilbert E (1959) Random graphs. Ann Math Stat 30(4):1141–1144

    Article  MATH  Google Scholar 

  • Gómez D, Montero J, Yáñez J (2006) A coloring fuzzy graph approach for image classification. Inf Sci 176(24):3645–3657

    Article  MATH  Google Scholar 

  • Jin Y, Hao J, Hamiez J (2014) A memetic algorithm for the minimum sum coloring problem. Comput Oper Res 43:318–327

    Article  MathSciNet  MATH  Google Scholar 

  • Keshavarz E (2016) Vertex-coloring of fuzzy graphs: a new approach. J Intell Fuzzy Syst 30(2):883–893

    Article  MATH  Google Scholar 

  • Liu B (2007) Uncertainty theory, 2nd edn. Springer, Berlin

    MATH  Google Scholar 

  • Liu B (2009) Some research problems in uncertainty theory. J Uncertain Syst 3(1):3–10

    Google Scholar 

  • Liu B (2012) Why is there a need for uncertainty theory. J Uncertain Syst 6(1):3–10

    Google Scholar 

  • Liu B (2014) Uncertain random graph and uncertain random network. J Uncertain Syst 8(1):3–12

    MathSciNet  Google Scholar 

  • Liu Y, Liu J, Wang K, Zhang H (2016) A theoretical extension on the operational law for monotone functions of uncertain variables. Soft Comput 20(11):4363–4376

    Article  MATH  Google Scholar 

  • Lü Z, Hao J (2010) A memetic algorithm for graph coloring. Eur J Oper Res 203(1):241–250

    Article  MathSciNet  MATH  Google Scholar 

  • Malaguti E, Monaci M, Toth P (2008) A metaheuristic approach for the vertex coloring problem. Inf J Comput 20(2):302–316

    Article  MathSciNet  MATH  Google Scholar 

  • Malaguti E, Toth P (2010) A survey on vertex coloring problems. Int Trans Oper Res 17(1):1–34

    Article  MathSciNet  MATH  Google Scholar 

  • Mann Z, Szajkó A (2013) Average-case complexity of backtrack search for coloring sparse random graphs. J Comput Syst Sci 79(8):1287–1301

    Article  MathSciNet  MATH  Google Scholar 

  • Morrison D, Sewell E, Jacobson S (2016) Solving the pricing problem in a branch-and-price algorithm for graph coloring using zero-suppressed binary decision diagrams. Inf J Comput 28(1):67–82

    Article  MathSciNet  MATH  Google Scholar 

  • Muñoz S, Ortuño M, Ramírez J, Yáñez J (2005) Coloring fuzzy graphs. Omega 33(3):211–221

    Article  Google Scholar 

  • Pramanik T, Samanta S, Sarkar B, Pal M (2017) Fuzzy \(\phi \)-tolerance competition graphs. Soft Comput 21(13):3723–3734

    Article  MATH  Google Scholar 

  • Robertson N, Sanders D, Seymour P, Thomas R (1997) The four-colour theorem. J Comb Theory Ser B 70(1):2–44

    Article  MathSciNet  MATH  Google 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 77–95

    Chapter  Google Scholar 

  • Rosyida I, Widodo Indrati C, Sugeng K (2015) A new approach in determining fuzzy chromatic number of a fuzzy graph. J Intell Fuzzy Syst 28(5):2331–2341

    Article  MATH  Google 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 1–8

  • 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

  • Samanta S, Pramanik T, Pal M (2016) Fuzzy colouring of fuzzy graphs. Afr Math 27(1):37–50

    Article  MathSciNet  MATH  Google Scholar 

  • Sommer C (2009) A note on coloring sparse random graphs. Discret Math 309(10):3381–3384

    Article  MathSciNet  MATH  Google Scholar 

  • Zadeh L (1965) Fuzzy sets. Inf Control 8(3):338–353

    Article  MATH  Google Scholar 

  • Zhang B, Peng J (2012) Euler index in uncertain graph. Appl Math Comput 218(20):10279–10288

    MathSciNet  MATH  Google Scholar 

  • Zhang B, Peng J (2013) Matching index of uncertain graph: concept and algorithm. Appl Comput Math 12(3):381–391

    MathSciNet  MATH  Google Scholar 

Download references

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.

Author information

Authors and Affiliations

  1. College of Management and Economics, Tianjin University, Tianjin, 300072, China

    Lin Chen

  2. Institute of Uncertain Systems, Huanggang Normal University, Huanggang, 438099, Hubei, China

    Jin Peng

  3. Department of Mathematical Sciences, University of Cincinnati, Cincinnati, OH, 45221-00025, USA

    Dan A. Ralescu

Authors
  1. Lin Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jin Peng
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Dan A. Ralescu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jin Peng.

Ethics declarations

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.

Additional information

Communicated by V. Loia.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chen, L., Peng, J. & Ralescu, D.A. Uncertain vertex coloring problem. Soft Comput 23, 1337–1346 (2019). https://doi.org/10.1007/s00500-017-2861-7

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00500-017-2861-7

Keywords

Search

Navigation