Range-valued fuzzy colouring of intuitionistic fuzzy graphs with application. (English) Zbl 07705706
Summary: Intuitionistic fuzzy graph, belonging to fuzzy graphs family has good capabilities when facing with problems that cannot be expressed by fuzzy graphs. Intuitionistic fuzzy graph can handle the vagueness connected with the incompatible and determinate information of any real-world problem, where fuzzy graphs may not succeed to bear satisfactory results. The previous restrictions in fuzzy graphs have led us to propose new definitions in intuitionistic fuzzy graphs. Colouring problem in graphs theory is one of the key issues that has many applications in computer science and social networks. Today, many researchers are trying to prove its application in medical sciences and psychology. Hence, in this paper, new concept of colouring of intuitionistic fuzzy graphs has been introduced. Also, some important terms like power cut graph of intuitionistic fuzzy graphs, range-valued fuzzy colour, chromatic number of an intuitionistic fuzzy graph colouring have been described. Some relevant results are proved. This technique is used to colour world political map mentioning the strength of relationship among the countries. Also, a new kind of traffic signal system has been proposed.
MSC:
| 68Q42 | Grammars and rewriting systems |
References:
| [1] | M. Akram, Level graphs of intuitionistic fuzzy graphs, Ann. Fuzzy Math. Inform., 16(2018), 55-70 · Zbl 1463.05449 |
| [2] | M. Akram and R. Akmal, Intuitionistic fuzzy graph structures, Kragujevac J. Math., 41(2017), 219-237. · Zbl 1468.05244 |
| [3] | M. Akram and R. Akmal, Operations on intuitionistic fuzzy graph structures, Fuzzy Inf. Eng., 8(2016), 389-410. |
| [4] | M. Akram, R. Akmal and N. Alshehri, On m-polar fuzzy graph structures, Springer Plus, (2016), 5:1448. |
| [5] | M. Akram, X. Peng and A. Sattar, A new decision-making model using complex intuition-istic fuzzy Hamacher aggregation operators, Soft Computing, 25(2021), 7059-7086. · Zbl 1498.91110 |
| [6] | M. Akram, G. Shahzadi and K.P. Shum, Operations on m-polar fuzzy r-uniform hypergraphs, Southeast Asian Bull. Math., 44(2020), 1-16. · Zbl 1449.05191 |
| [7] | F. S. Al-Anzia, Y. N. Sotskov, A. Allahverdi and G. V. Andreev, Using mixed graph coloring to minimize total completion time in job shop scheduling, Appl. Math. Compu, 182(2006), 1137-1148. · Zbl 1114.90033 |
| [8] | K. T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 20(1986), 87-96. · Zbl 0631.03040 |
| [9] | K. T. Atanassov, Intuitionistic Fuzzy Sets. Theory and Applications. Studies in Fuzziness and Soft Computing, 35. Physica-Verlag, Heidelberg, 1999. · Zbl 0939.03057 |
| [10] | L. S. Bershtein and A. V. Bozhenuk, Fuzzy coloring for fuzzy graphs, IEEE Int. Fuzzy Syst. Conf, 3(2001), 1101-1103. |
| [11] | K. R. Bhutani and A. Rosenfeld, Strong arcs in fuzzy graphs, Inf. Sci, 152(2003), 319-322. · Zbl 1040.03518 |
| [12] | R. A. Borzooei and H. Rashmanlou, Semi global domination sets in vague graphs with application, Journal of Intelligent and Fuzzy Systems, 30(2016), 3645-3652. · Zbl 1361.05093 |
| [13] | R. A. Borzooei and H. Rashmanlou, Cayley interval-valued fuzzy graphs, UPB Scientific Bulletin, Series A: Applied Mathematics and Physics, 78(2016), 83-94. · Zbl 1513.05314 |
| [14] | R. A. Borzooei and H. Rashmanlou, Domination in vague graphs and its applications, J. Intell. Fuzzy Systems, 29(2015), 1933-1940. · Zbl 1362.05094 |
| [15] | R. A. Borzooei and H. Rashmanlou, Ring sum in product intuitionistic fuzzy graphs, J. Adv. Res. Pure Math., 7(2015), 16-31. |
| [16] | N. Christofides, Graph Theory: An Algorithmic Approach, Academic Press, (1975). · Zbl 0321.94011 |
| [17] | M. Gamache, A. Hertz and J. O. Ouellet, A graph coloring model for a feasibility problem in monthly crew scheduling with preferential bidding, Comput. Operations Res, 34(2007), 2384-2395. · Zbl 1144.90498 |
| [18] | G. J. Klir and B. Yuan, Fuzzy Sets and Fuzzy Logic: Theory and Application, PHI, (1997). |
| [19] | L. T. Koczy, Fuzzy graphs in the evaluation and optimization of networks, Fuzzy Sets and Systems, 46(1992), 307-319. · Zbl 0761.05092 |
| [20] | J. Korst, E. Aartsas, J. K. Lenstra and J. Wessels, Periodic assignment and graph colouring, Discrete Appl. Math., 51(1994), 291-305. · Zbl 0807.05030 |
| [21] | K. K. Krishna, H. Rashmanlou, A. A. Talebi and F. Mofidnakhaei, Regularity of cubic graph with application, J. Indones. Math. Soc., 25(2019), 1-15. · Zbl 1451.05197 |
| [22] | K. K. Krishna, Y. Talebi, H. Rashmanlou and A. A. Talebi, New concept of cubic graph with application, Journal of Multiplevalued Logic and Soft Computing, 33(2019), 135-154. · Zbl 1452.05149 |
| [23] | R. Lewis and J. Thompson, On the application of graph colouring techniques in round-robin sports scheduling, Comput. Oper. Res., 38(2011), 190-204. · Zbl 1231.90206 |
| [24] | C. L. Liu, Introduction to Combinatorial Mathematics, McGraw-Hill, (1968). · Zbl 0188.03801 |
| [25] | J. N. Mordeson and C. S. Peng, Operations on fuzzy graphs, Inform. Sci., 79(1994), 159-170. · Zbl 0804.05069 |
| [26] | J. N. Mordeson and P. S. Nair, Fuzzy Graphs and Hypergraphs, Physica Verlag, 2000. · Zbl 0947.05082 |
| [27] | S. Munoz, M. T. Ortuo, J. Ramrez and J. Yez, Coloring fuzzy graphs, Omega, 33(2005), 211-221. |
| [28] | S. K. Pal and S. S. Sarma, Graph coloring approach for hiding of information, Procedia Technol, 4(2012), 272-227. |
| [29] | R. Parvathi and M. G. Karunambigai, Intuitionistic fuzzy graphs, Journal of computational Intelligence: Theory and Applications, (2006), 139-150. |
| [30] | H. Rashmanlou, R. A. Borzooei, M. Shoaib, Y. Talebi, M. Taheri and F. Mofidnakhaei, New way for finding shortest path problem in a network, Journal of Multiple-Valued Logic & Soft Computing, 34(2020), 451-460. |
| [31] | H. Rashmanlou, Y. B. Jun and R. A. Borzooei, More results on highly irregular bipolar fuzzy graphs, Ann. Fuzzy Math. Inform., 8(2014), 149-168. · Zbl 1306.05200 |
| [32] | H. Rashmanlou, S. Samanta, M. Pal and R. A. Borzooei, Intuitionistic fuzzy graphs with categorical properties, Fuzzy Inf. Eng., 7(2015), 317-334. |
| [33] | H. Rashmanlou, S. Samanta, M. Pal and R. A. Borzooei, Product of bipolar fuzzy graphs and their degree, Int. J. Gen. Syst., 45(2016), 1-14. · Zbl 1343.05125 |
| [34] | H. Rashmanlou, S. Samanta, M. Pal and R. A. Borzooei, Bipolar fuzzy graphs with Categorical prop-erties, International Journal of Computational Intelligence Systems, 8(2015), 808-818. |
| [35] | H. Rashmanlou, S. Samanta, M. Pal and R. A. Borzooei, A study on bipolar fuzzy graphs, J. Intell. Fuzzy Systems, 28(2015), 571-580. · Zbl 1351.05188 |
| [36] | A. Rosenfeld, Fuzzy graphs, Fuzzy Sets and Their Applications (L.A. Zadeh, K.S. Fu, M. Shimura, Eds.) Academic Press, New York (1975), 77-95. · Zbl 0315.05131 |
| [37] | S. Sahoo, M. Pal, H. Rashmanlou and R.A. Borzooei, Covering and paired domination in intuitionistic fuzzy graphs, J. Intell. Fuzzy Systems, 33(2017), 4007-4015. |
| [38] | S. Samanta and M. Pal, Fuzzy k-competition graphs and p-competition fuzzy graphs, Fuzzy Inf. Eng., 5(2013), 191-204. · Zbl 1430.05101 |
| [39] | S. Satyanarayana, Range-valued fuzzy colouring of interval-valued fuzzy graphs, Natural Science and Engineering, 18(2016), 169-177. |
| [40] | S. Satyanarayana Ch. Ramprasad and N. Srinivasarao, A study on interval-valued fuzzy graphs, Com-puter Science and Telecommunications, 4(2016), 13 pages. |
| [41] | S. Shahzadi and M. Akram, Intuitionistic fuzzy soft graphs with applications, J. Appl. Math. Comput., 55(2017), 369-392. · Zbl 1388.05154 |
| [42] | S. Shahzadi and M. Akram, Coloring of bifuzzy graphs, Ital. J. Pure Appl. Math., 36(2016), 429-444. · Zbl 1383.05267 |
| [43] | A. A. Talebi and W. A. Dudek, Operations on level graphs of bipolar fuzzy graphs, Bul. Acad. Ştiinţe Repub. Mold. Mat., 2(2016), 107-124. · Zbl 1355.05205 |
| [44] | A. A. Talebi, M. Ghassemi and H. Rashmanlou, New concepts of irregular-intuitionistic fuzzy graphs with applications, An. Univ. Craiova Ser. Mat. Inform., 47(2020), 226-243. · Zbl 1488.05424 |
| [45] | A. A. Talebi, M. Ghassemi, H. Rashmanlou and S. Broumi, Novel properties of edge irregular single valued neutrosophic graphs, Neutrosophic Sets and Systems(NSS), 43(2021), 255-279. |
| [46] | A. A. Talebi, H. Rashmanlou, M. Ghassemi, New concepts of strongly edge irregular interval-valued neutrosophic graphs, Nova Science Publishers, Inc., July 2020, In Press. · Zbl 1488.05424 |
| [47] | A. A. Talebi, J. Kacprzyk, H. Rashmanlou and S.H. Sadati, A new concept of an intuitionistic fuzzy graph with applications, J. Mult.-Valued Logic Soft Comput., 35(2020), 431-454. · Zbl 1487.05220 |
| [48] | Y. Talebi and H. Rashmanlou, New concepts of domination sets in vague graphs with applications, Int. J. Comput. Sci. Math., 10(2019), 375-389. · Zbl 1453.05109 |
| [49] | S. Vimala and J. S. Sathya, Connected point set domination of fuzzy graphs, International Journal of Mathematics and Soft Computing, 2(2012), 75-78. |
| [50] | V. Yegnanarayanan, Graph colourings and partitions, Theoret. Comput. Sci., 263(2001), 59-74. · Zbl 0979.68065 |
| [51] | L. A. Zadeh, Fuzzy Sets, Information and Control, 8(1965), 338-353 · Zbl 0139.24606 |
| [52] | L. A. Zadeh, Similarity relations and fuzzy orderings, Information Sci., 3(1971), 177-200. · Zbl 0218.02058 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
