Coloring picture fuzzy graphs through their cuts and its computation 
^{(2) } Suryono Suryono (Department of Doctor of Information System, Universitas Diponegoro, Indonesia)
^{*}corresponding author
AbstractIn a fuzzy set (FS), there is a concept of alphacuts of the FS for alpha in [0,1]. Further, this concept was extended into (alpha,delta)cuts in an intuitionistic fuzzy set (IFS) for delta in [0,1]. One of the expansions of FS and IFS is the picture fuzzy set (PFS). Hence, the concept of (alpha,delta)cuts was developed into (alpha,delta,beta)cuts in a PFS where beta is an element of [0,1]. Since a picture fuzzy graph (PFG) consists of picture fuzzy vertex or edge sets or both of them, we have an idea to construct the notion of the (alpha,delta,beta)cuts in a PFG. The steps used in this paper are developing theories and algorithms. The objectives in this research are to construct the concept of (alpha,delta,beta)cuts in picture fuzzy graphs (PFGs), to construct the (alpha,delta,beta)cuts coloring of PFGs, and to design an algorithm for finding the cut chromatic numbers of PFGs. The first result is a definition of the (alpha,delta,beta)cut in picture fuzzy graphs (PFGs) where (alpha,delta,beta) are elements of a level set of the PFGs. Further, some properties of the cuts are proved. The second result is a concept of PFG coloring and the chromatic number of PFG based on the cuts. The third result is an algorithm to find the cuts and the chromatic numbers of PFGs. Finally, an evaluation of the algorithm is done through Matlab programming. This research could be used to solve some problems related to theories and applications of PFGs.
KeywordsPicture fuzzy graph; Coloring graph; Cuts; Chromatic number; Intuitionistic fuzzy set

DOIhttps://doi.org/10.26555/ijain.v7i1.612 
Article metricsAbstract views : 572  PDF views : 121 
Cite 
Full TextDownload 
References
[1] S. Mathew, J. N. Mordeson, and D. S. Malik, “Fuzzy Graph Theory,” in Studies in Fuzziness and Soft Computing, Second., Switzerland: Springer Nature, 2018, pp. 13–76, doi: 10.1007/9783319714073.
[2] K. T. Atanassov, “On intuitionistic fuzzy graphs and intuitionistic fuzzy relations,” in Proceedings of the VI IFSA World Congress, 1995, pp. 551–554. Available at: Google Scholar.
[3] B. C. Cuong and V. Kreinovich, “Picture fuzzy sets  A new concept for computational intelligence problems,” in 2013 3rd World Congress on Information and Communication Technologies, WICT, 2013, pp. 1–6, doi: 10.1109/WICT.2013.7113099
[4] B. C. Cuong, “Picture fuzzy sets,” J. Comput. Sci. Cybern., vol. 30, no. 4, pp. 409–420, 2015, doi: 10.15625/18139663/30/4/5032.
[5] P. Dutta and S. Ganju, “Some aspects of picture fuzzy set,” Trans. A. Razmadze Math. Inst., vol. 172, no. 2, pp. 164–175, 2018, doi: 10.1016/j.trmi.2017.10.006.
[6] B. Davvaz, N. Jan, T. Mahmood, and K. Ullah, “Intuitionistic Fuzzy Graphs of n Type with Applications,” J. Intell. Fuzzy Syst., vol. 36, no. 4, pp. 3923–3932, 2019, doi: 10.3233/JIFS181123
[7] M. Pal, S. Samanta, and G. Ghorai, “Intuitionistic Fuzzy Graphs,” 2020, 1st ed., pp. 225–274, doi: https://doi.org/10.1007/9789811588037_9, doi: 10.1007/9789811588037_9.
[8] T. Alhawary, T. Mahmood, N. Jan, and K. Ullah, “On Intuitionistic Fuzzy Graphs and Some Operations on Picture Fuzzy Graphs On Intuitionistic Fuzzy Graphs and Some Operations on,” Ital. J. Pure Appl. Math. ·, vol. In Press, pp. 1–15, 2018, available at: Google Scholar.
[9] P. Dutta and K. Saikia, “Some aspects of Equivalence Picture Fuzzy Relation Palash,” Adv. Model. Anal. A, vol. 172, no. 2, pp. 164–175, 2018, doi: 10.1016/j.trmi.2017.10.006.
[10] C. Zuo, A. Pal, and A. Dey, “New concepts of picture fuzzy graphs with application,” Mathematics, vol. 7, no. 5, pp. 1–18, 2019, doi: 10.3390/math7050470.
[11] W. Xiao, A. Dey, and L. H. Son, “A study on regular picture fuzzy graph with applications in communication networks,” J. Intell. Fuzzy Syst., vol. 39, no. 3, pp. 3633–3645, 2020, doi: 10.3233/JIFS191913.
[12] M. Ismayil A, R. U. Rehman A, and R. Tejaskumar, “Edge Domination in Picture Fuzzy Graphs,” Int. J. Comput. Eng. Res., vol. 09, no. 08, pp. 39–45, 2019. Available at: Google Scholar.
[13] S. Jayalakshmi and D. Vidhya, “On Direct Sum of Two Picture Fuzzy Graph,” in AIP Conf. Proc., 2020, vol. 2277, no. 090004, pp. 1–7, doi: 10.1063/5.0025300.
[14] A. Talebi, H. Rashmanlou, and S. H. Sadati, “Several Notions of Energy in Picture Fuzzy Graphs,” in 4th International Conference on Combinatorics, Criptography, Computer Science and Computing (I4C), 2019, pp. 200–228, available at: Google Scholar.
[15] L. T. Koczy, N. Jan, T. Mahmood, and K. Ullah, “Analysis of social networks and WiFi networks by using the concept of picture fuzzy graphs,” Soft Comput., vol. 24, no. 21, pp. 16551–16563, 2020, doi: 10.1007/s00500020049599.
[16] K. Mohanta, A. Dey, and A. Pal, “A study on picture dombi fuzzy graph,” Decis. Manag. andEngineering, vol. 3, no. 2, pp. 119–130, 2020, doi: 10.31181/dmame2003119m.
[17] S. Das and G. Ghorai, “Analysis of the effect of medicines over bacteria based on competition graphs with picture fuzzy environment,” Comput. Appl. Math., vol. 39, no. 3, pp. 1–21, 2020, doi: 10.1007/s40314020011966.
[18] S. Das and G. Ghorai, “Analysis of Road Map Design Based on Multigraph with Picture Fuzzy Information,” Int. J. Appl. Comput. Math., vol. 6, no. 3, pp. 1–17, 2020, doi: 10.1007/s40819020008163.
[19] S. Das, G. Ghorai, and M. Pal, Certain Competition Graphs Based on Picture Fuzzy Environment with Applications, 2020, doi: 10.1007/s10462020099235, doi: 10.1007/s10462020099235.
[20] S. Das, G. Ghorai, and M. Pal, “Genus of Graphs Under Picture Fuzzy Environment with Applications,” J. Ambient. Intell. Humaniz. Comput., 2021, doi: 10.1007/s1265202002887y.
[21] M. Sitara, M. Akram, and M. Riaz, “DecisionMaking Analysis Based on qRung Picture Fuzzy Graph Structures,” J Appl Math Comput ., 2021, doi: 10.1007/s1219002001471z.
[22] P. Mani, B. Vasudevan, and M. Sivaraman, “Shortest path algorithm of a network via picture fuzzy digraphs and its application,” Mater. Today Proc., 2021, doi: 10.1016/j.matpr.2020.12.006.
[23] M. Pal, S. Samanta, and G. Ghorai, “Coloring of Fuzzy Graph,” 2020, 1st ed., pp. 175–193, doi: 10.1007/9789811588037_7.
[24] I. Rosyida, Widodo, C. R. Indrati, D. Indriati, and Nurhaida, “Fuzzy Chromatic Number of Union of Fuzzy Graphs: An Algorithm, Properties and its Application,” Fuzzy Sets Syst., vol. 384, pp. 115–131, 2020, doi: 10.1016/j.fss.2019.04.028.
[25] I. Rosyida, J. Peng, L. Chen, W. Widodo, C. R. Indrati, and K. A. Sugeng, “An Uncertain Chromatic Number of an Uncertain Graph Based on αCut Coloring,” Fuzzy Optim Decis Mak., vol. 17, pp. 103–123, 2018, doi: 10.1007/s107000169260x.
[26] I. Rosyida, Widodo, C. R. Indrati, and D. Indriati, “On construction of fuzzy chromatic number of cartesian product of path and other fuzzy graphs,” J. Intell. Fuzzy Syst., vol. 39, no. 1, pp. 1073–1080, 2020, doi: 10.3233/JIFS191982.
[27] I. Rosyida, Widodo, C. R. Indrati, and K. A. Sugeng, “A New Approach for Determining Fuzzy Chromatic Number of Fuzzy Graph”, J. Intell. Fuzzy Syst., vol. 28, no. 5, pp. 2331–2341, 2015, doi: 10.3233/IFS141521.
[28] S. Ismail Mohideen and M. . Rifayathali, “Coloring of intuitionistic fuzzy graph using (alpha,beta)cuts.,” Int. Res. J. Math. Eng. IT, vol. 2, no. 12, pp. 14–26, 2015. Available at: Google Scholar.
[29] A. Prasanna, M. . Rifayathali, and S. Ismail Mohideen, “Strong intuitionistic fuzzy graphs,” Int. J. Latest Eng. Res. Appl., vol. 02, no. 08, pp. 163–169, 2017, doi: 10.1007/s4147801801029.
[30] M. Ismayil, “Domination in Picture Fuzzy Graphs,” in 5th International Conference on Mathematical Methods and Computation (ICOMAC  2019), 2019, no. February, 2021, pp. 205–210. Available at: Google Scholar.
This work is licensed under a Creative Commons AttributionShareAlike 4.0 International License.
___________________________________________________________
International Journal of Advances in Intelligent Informatics
ISSN 24426571 (print)  25483161 (online)
Organized by Informatics Department  Universitas Ahmad Dahlan, and ASCEE Computer Society
Published by Universitas Ahmad Dahlan
W: http://ijain.org
E: ijain@uad.ac.id (paper handling issues)
info@ijain.org, andri.pranolo.id@ieee.org (publication issues)
This work is licensed under a Creative Commons AttributionShareAlike 4.0