Abstract
In this research, a new mathematical integer programming model is presented for the graph labeling problem of quadratic graphs. The advantages of this model are linearity and the existence of an objective function. Furthermore, two constraint programming models and a meta-heuristics algorithm are also developed to generate feasible graceful labeling and \(\alpha \)-labeling for special classes of quadratic graphs. Experimental results on large sizes of graphs from the literature show the efficiency of the proposed model and approach.
Similar content being viewed by others
References
Eshghi, K.: Existence and construction of-labeling of 2-regular graphs with three components. PhD diss., Ph. D. thesis, University of Toronto (1997)
Gallian, J.A.: A dynamic survey of graph labeling. Electron. J. Comb., 20th edn. 1–415 (2017)
Eshghi, K.: Extension of \(\alpha \)-labelings of quadratic graphs. Int. J. Math. Math. Sci. 2004(11), 571–578 (2004)
Eshghi, K., Salarrezaie, M.: An integer programming model and a Tabu search algorithm to generate \(\alpha \)-labeling of special classes of quadratic graph. Iran. J. Oper. Res. (to appear)
Eshghi, K., Salarrezaie, M.: Existence and construction of \(\alpha \)-labeling for quadratic graph \(Q(7,4k)\) and its extensions. Int. J. Graph. Theory Appl. (IJGTA) (to appear)
Rosa, A.: On certain valuations of the vertices of a graph. In: Theory of Graphs, International Symposium, Rome, pp. 349–355 (1966)
Abrham, J., Kotzig, A.: On the missing value in a graceful numbering of a 2-regular graph. Congr. Numer. 65(0988), 261–266 (1988)
Kotzig, A.: Decompositions of complete graphs into isomorphic cubes. J. Comb. Theory Ser. B 31(3), 292–296 (1981)
Lakshmi, D.R., Vangipuram, S.: A note on the graceful numbering of a class of trees. Def. Sci. J. 35, 65–70 (1985)
Salarrezaei, M.: Mathematical programming models for graceful labeling problem of 2-regular graphs. Master of Science thesis, Sharif University of Technology (2012)
Eshghi, K., Azimi, P.: Applications of mathematical programming in graceful labeling of graphs. J. Appl. Math. 1(1), 1–8 (2004)
Eshghi, K., Azimi, P.: An algorithm for finding a feasible solution of graph labeling problems. Utilitas Mathematica 72, 163–174 (2007)
Redl, T.A.: Graceful graphs and graceful labelings: two mathematical programming formulations and some other new results. Congressus Numerantium 164, 17–32 (2003)
Smith, B.M., Puget, J.-F.: Constraint models for graceful graphs. Constraints 15(1), 64–92 (2010)
Mahmoudzadeh, H., Eshghi, K.: Metaheuristic approach to the graceful labeling problem. Int. J. Appl. Metaheuristic Comput. IJAMC 1, 42–57 (2010)
Shaabani, H., Kamalabadi, I.N.: An efficient population-based simulated annealing algorithm for the multi-product multi-retailer perishable inventory routing problem. Comput. Ind. Eng. 99, 189–201 (2016)
Kirkpatrick, S., Gelatt, C.D., Vecchi, M.P.: Optimization by simulated annealing. Science 220(4598), 671–680 (1983)
Goldberg, D.E., Kalyanmoy, D.: A comparative analysis of selection schemes used in genetic algorithms. Found. Genet. Algorithms 1, 69–93 (1991)
Author information
Authors and Affiliations
Corresponding author
Electronic supplementary material
Below is the link to the electronic supplementary material.
Rights and permissions
About this article
Cite this article
Amini, M., Eshghi, K. Constraint programming models and population-based simulated annealing algorithm for finding graceful and \(\alpha \)-labeling of quadratic graphs. Iran J Comput Sci 1, 155–164 (2018). https://doi.org/10.1007/s42044-018-0012-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s42044-018-0012-7