Recent Trends and Developments in Graph Coloring

  • Malti Baghel
  • Shikha Agrawal
  • Sanjay Silakari
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 199)


This paper is intended to give review of various heuristics and metaheuristics methods to graph coloring problem. The graph coloring problem is one of the combinatorial optimization problems used widely. It is a fundamental and significant problem in scientific computation and engineering design. The graph coloring problem is an NP-hard problem and can be explained as given an undirected graph, one has to find the least number of colors for coloring the vertices of the graph such that the two adjacent vertices must have different color. The minimum number of colors needed to color a graph is called its chromatic number. In this paper, a brief survey of various methods is given to solve graph coloring problem. Basically we have categorized it into three parts namely heuristic method, metaheuristic methods and hybrid methods. This paper surveys and analyzes various methods with an emphasis on recent developments.


Ant colony optimization Genetic algorithm particle swarm optimization simulated annealing Tabu search 


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  1. 1.
    Hertz, A., de Werra, D.: Using tabu search techniques for graph coloring. Comput. 39(4), 345–351 (1987)CrossRefMATHGoogle Scholar
  2. 2.
    Lim, A., Wang, F.: Meta-heuristics for robust graph coloring problem. In: Proceedings of 16th IEEE International Conference on Tools with Artificial Intelligence, Florida, pp. 514–518 (2004)Google Scholar
  3. 3.
    Ray, B., Pal, A.J., Bhattacharyya, D., Kim, T.H.: An Efficient GA with Multipoint Guided Mutation for Graph Coloring Problems. Int. J. Signal Process. Image Process. and Pattern Recognit. 3(2), 51–58 (2010)Google Scholar
  4. 4.
    Brelaz, D.: New methods to color the vertices of a graph. Commun. ACM. 22, 251–256 (1979)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Avanthay, C., Hertz, A., Zufferey, N.: A variable neighborhood search for Graph coloring. Eur. J. Oper. Res. 151(2), 379–388 (2003)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Fleurent, C., Ferland, J.A.: Genetic and hybrid algorithms for graph coloring. Ann. Oper. Res. 63(3), 437–461 (1996)CrossRefMATHGoogle Scholar
  7. 7.
    Chiarandini, M., Stutzle, T.: An application of iterated local search to graph coloring. In: Johnson, D.S., Mehrotra, A., Trick, M. (eds.) Proc. of the Computational Symposium on Graph Coloring and its Generalizations, Ithaca, New York, USA, pp. 112–125 (2002)Google Scholar
  8. 8.
    Cui, G., Qin, L., Liu, S., Wang, Y., Zhang, X., Cao, X.: Modified PSO algorithm for solving planar graph coloring problem. Progress Nat. Sci. 18, 353–357 (2008)CrossRefGoogle Scholar
  9. 9.
    Costa, D., Hertz, A.: Ants Can Color Graphs. J. Oper. Res. Soc. 48, 295–305 (1997)MATHGoogle Scholar
  10. 10.
    Johnson, D.S., Aragon, C.R., McGeoch, L.A., Schevon, C.: Optimization by simulated annealing:an experimental evaluation; part II, graph coloring and number partitioning. Oper. Res. 39(3), 378–406 (1991)CrossRefMATHGoogle Scholar
  11. 11.
    Porumbel, D.C., Hao, J.-K., Kuntz, P.: Position-Guided Tabu Search Algorithm for the Graph Coloring Problem. In: Stützle, T. (ed.) LION 3. LNCS, vol. 5851, pp. 148–162. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  12. 12.
    Costa, D., Hertz, A., Dubuis, C.: Embedding a sequential procedure within an evolutionary algorithm for coloring problems in graphs. J. Heuristics 1, 105–128 (1995)CrossRefMATHGoogle Scholar
  13. 13.
    Dorigo, M., Maniezzo, V., Colorni, A.: Positive feedback as a search strategy. Technical Report 91-016, Politecnico di Milano, Italy (1991)Google Scholar
  14. 14.
    Dorne, R., Hao, J.K.: Tabu Search for graph coloring, T-coloring and Set T-colorings. In: Osman, I.H., et al. (eds.) Metaheuristics 1998: Theory and Applications. ch. 3. Kluver Academic Publishers (1998)Google Scholar
  15. 15.
    Salari, E., Eshghi, K.: An ACO Algorithm for the Graph Coloring Problem. Int. J. Contemp. Math. Sci. 3, 293–304 (2008)MATHMathSciNetGoogle Scholar
  16. 16.
    Erfani, M.: A modified PSO with fuzzy inference system for solving the planar graph coloring problem. Masters thesis, Universiti Teknologi Malaysia, Faculty of Computer Science and Information System (2010)Google Scholar
  17. 17.
    Glover, F.: Future paths for integer programming and links to artificial intelligence. Comput. Oper. Res. 13, 533–549 (1986)CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Leighton, F.T.: A graph coloring algorithm for large scheduling problems. J. Res. Natl. Bur. Stand. 84(6), 489–505 (1979)CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Garey, R., Johnson, D.S.: A guide to the theory of NP–completeness. Computers and intractability. W. H. Freeman, New York (1979)MATHGoogle Scholar
  20. 20.
    Gendron, B., Hertz, A., St-Louis, P.: On edge orienting methods for graph coloring. J. of Comb. Optim. 13(2), 163–178 (2007)CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    Hertz, A., Plumettaz, M., Zufferey, N.: Variable space search for graph coloring. Discret. Appl. Math. 156(13), 2551–2560 (2008)CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Hsu, L., Horng, S., Fan, P.: Mtpso algorithm for solving planar graph coloring problem. Expert Syst. Appl. 38, 5525–5531 (2011)CrossRefGoogle Scholar
  23. 23.
    Ayanegui, H., Chavez-Aragon, A.: A complete algorithm to solve the graph-coloring problem. In: Fifth Latin American Workshop on Non-Monotonic Reasoning, LANMR, pp. 107–117 (2009)Google Scholar
  24. 24.
    Blochliger, I., Zufferey, N.: A graph coloring heuristic using partial solutions and a reactive tabu scheme. Comput. Oper. Res. 35(3), 960–975 (2008)CrossRefMathSciNetGoogle Scholar
  25. 25.
    Holland, J.H.: Adaption in natural and artificial systems. The University of Michigan Press, Ann Harbor (1975)MATHGoogle Scholar
  26. 26.
    Qin, J., Yin, Y.-X., Ban, X.-J.: Hybrid discrete particle swarm optimization for graph coloring problem. J. Comput. 6, 1175–1182 (2011)Google Scholar
  27. 27.
    Kennedy, J., Eberhart, R.C.: Particle swarm optimization. In: Proc. of IEEE Int. Conf. Neural Netw., Piscataway, NJ, USA, pp. 1942–1948 (1995)Google Scholar
  28. 28.
    Johnson, D.S.: Approximation algorithms for combinatorial problems. J. Comp. Syst. Sci. 9, 256–278 (1974)CrossRefMATHGoogle Scholar
  29. 29.
    Kirkpatrick, S., Gelatt, C.D., Vecchi, M.P.: Optimization by simulated annealing. Sci. 220, 671–680 (1983)CrossRefMATHMathSciNetGoogle Scholar
  30. 30.
    Davis, L.: Order-based genetic algorithms and the graph coloring problem. In: Handbook of Genetic Algorithms, pp. 72–90 (1991)Google Scholar
  31. 31.
    Chams, M., Hertz, A., Werra, D.: Some experiments with simulated annealing for coloring graphs. Eur. J. of Oper. Res. 32(2), 260–266 (1987)CrossRefMATHGoogle Scholar
  32. 32.
    Plumettaz, M., Schindl, D., Zufferey, N.: Ant Local Search and its effcient adaptation to graph colouring. Journal of Operational Research Society 61(5), 819–826 (2010)CrossRefMATHGoogle Scholar
  33. 33.
    Matula, D.W., Marble, G., Isaacson, D.: Graph coloring algorithms. In: Graph Theory and Computing, pp. 109–122. Academic Press, New York (1972)Google Scholar
  34. 34.
    Mladenovic, N., Hansen, P.: Variable Neighborhood Search. Comput. Oper. Res. 24, 1097–1100 (1997)CrossRefMATHMathSciNetGoogle Scholar
  35. 35.
    Galinier, P., Hao, J.K.: Hybrid evolutionary algorithms for graph coloring. J. Comb Optim. 3(4), 379–397 (1999)CrossRefMATHMathSciNetGoogle Scholar
  36. 36.
    Chalupa, D.: Population-based and learning-based metaheuristic algorithms for the graph coloring problem. In: Krasnogor, N., Lanzi, P.L. (eds.) GECCO, pp. 465–472. ACM (2011)Google Scholar
  37. 37.
    Sivanandam, S.N., Sumathi, S., Hamsapriya, T.: A hybrid parallel genetic algorithm approach for graph coloring. Int. J. Knowl. Based Intel. Eng. Syst. 9, 249–259 (2005)Google Scholar
  38. 38.
    Lukasik, S., Kokosinski, Z., Swieton, G.: Parallel Simulated Annealing Algorithm for Graph Coloring Problem. Parallel Process. Appl. Math., 229–238 (2007)Google Scholar
  39. 39.
    Titiloye, O., Crispin, A.: Quantum annealing of the graph coloring problem. Discret. Optim. 8(2), 376–384 (2011)CrossRefMATHMathSciNetGoogle Scholar
  40. 40.
    Trick, M.A., Yildiz, H.: A Large Neighborhood Search Heuristic for Graph Coloring. In: Van Hentenryck, P., Wolsey, L.A. (eds.) CPAIOR 2007. LNCS, vol. 4510, pp. 346–360. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  41. 41.
    Welsh, D.J., Powell, M.B.: An upper bound for the chromatic number of a graph and its application to timetabling problem. Comp. J. 10, 85–86 (1967)CrossRefMATHGoogle Scholar
  42. 42.
    Lu, Z., Hao, J.-K.: A memetic algorithm for graph coloring. Eur. J. Oper. Res. 203(1), 241–250 (2010)CrossRefMathSciNetGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.UIT, RGPVBhopalIndia

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