## Journal of Applied Mathematics

### Rainbow Connectivity Using a Rank Genetic Algorithm: Moore Cages with Girth Six

#### Abstract

A rainbow $t$-coloring of a $t$-connected graph $G$ is an edge coloring such that for any two distinct vertices $u$ and $v$ of $G$ there are at least $t$ internally vertex-disjoint rainbow $(u,v)$-paths. In this work, we apply a Rank Genetic Algorithm to search for rainbow $t$-colorings of the family of Moore cages with girth six $(t;\mathrm{6})$-cages. We found that an upper bound in the number of colors needed to produce a rainbow 4-coloring of a $(\mathrm{4};\mathrm{6})$-cage is 7, improving the one currently known, which is 13. The computation of the minimum number of colors of a rainbow coloring is known to be NP-Hard and the Rank Genetic Algorithm showed good behavior finding rainbow $t$-colorings with a small number of colors.

#### Article information

Source
J. Appl. Math., Volume 2019 (2019), Article ID 4073905, 7 pages.

Dates
Received: 12 September 2018
Accepted: 29 January 2019
First available in Project Euclid: 16 May 2019

Permanent link to this document
https://projecteuclid.org/euclid.jam/1557972261

Digital Object Identifier
doi:10.1155/2019/4073905

Mathematical Reviews number (MathSciNet)
MR3924131

#### Citation

Cervantes-Ojeda, J.; Gómez-Fuentes, M.; González-Moreno, D.; Olsen, M. Rainbow Connectivity Using a Rank Genetic Algorithm: Moore Cages with Girth Six. J. Appl. Math. 2019 (2019), Article ID 4073905, 7 pages. doi:10.1155/2019/4073905. https://projecteuclid.org/euclid.jam/1557972261

#### References

• D. Dasgupta and Z. Michalewicz, Evolutionary Algorithms in Engineering Applications, Springer Science and Business Media, 2013.
• K. A. De Jong and W. M. Spears, “Using genetic algorithms to solve NP-complete problems,” International Computer Games Association, pp. 124–132, 1989.
• S. Jakobs, “On genetic algorithms for the packing of polygons,” European Journal of Operational Research, vol. 88, no. 1, pp. 165–181, 1996.
• A. Pourrajabian, R. Ebrahimi, M. Mirzaei, and M. Shams, “Applying genetic algorithms for solving nonlinear algebraic equations,” Applied Mathematics and Computation, vol. 219, no. 24, pp. 11483–11494, 2013.
• J. Cervantes and C. R. Stephens, “Limitations of existing mutation rate heuristics and how a rank GA overcomes them,” IEEE Transactions on Evolutionary Computation, vol. 13, no. 2, pp. 369–397, 2009.
• S. Chakraborty, E. Fischer, A. Matsliah, and R. Yuster, “Hardness and algorithms for rainbow connectivity,” in Proceedings of the 26th International Symposium on Theoretical Aspects of Computer Science STACS, pp. 243–254, 2009, Also, see Journal of Combinatorial Optimization, vol. 21, pp. 330–347, 2011.
• K. Menger, “Zur allgemeinen Kurventheorie,” Fundamenta Mathematicae, vol. 10, pp. 96–115, 1927.
• G. Chartrand, G. L. Johns, K. A. McKeon, and P. Zhang, “The rainbow connectivity of a graph,” Networks. An International Journal, vol. 54, no. 2, pp. 75–81, 2009.
• G. Chartrand, G. L. Johns, K. A. McKeon, and P. Zhang, “Rainbow connection in graphs,” Mathematica Bohemica, vol. 133, no. 1, pp. 85–98, 2008.
• A. Ericksen, “A matter of security,” Graduating Engineer and Computer Careers, pp. 24–28, 2007.
• V. B. Le and Z. Tuza, “Finding optimal rainbow connection is hard,” 2009.
• X. Li, Y. Shi, and Y. Sun, “Rainbow connections of graphs: a survey,” Graphs and Combinatorics, vol. 29, no. 1, pp. 1–38, 2013.
• X. Li and Y. Sun, Rainbow Connections of Graphs, Springer, London, UK, 2013.
• G. Exoo and R. Jajcay, “Dynamic cage survey,” The Electronic Journal of Combinatorics, vol. DS16, 2013.
• M. Miller and J. Siran, “Moore graphs and beyond: a survey of the degree/diameter problem,” The Electronic Journal of Combinatorics - Dynamic Surveys, vol. 14, 2005.
• E. Bannai and T. Ito, “On finite Moore graphs,” Journal of the Faculty of Science. University of Tokyo, vol. 20, pp. 191–208, 1973.
• R. M. Damerell, “On Moore graphs,” Mathematical Proceedings of the Cambridge Philosophical Society, vol. 74, no. 2, pp. 227–236, 1973.
• A. J. Hoffman and R. R. Singleton, “On Moore graphs with diameters 2 and 3,” International Business Machines Journal of Research and Development, vol. 4, pp. 497–504, 1960.
• X. Marcote, C. Balbuena, and I. Pelayo, “On the connectivity of cages with girth five, six and eight,” Discrete Mathematics, vol. 307, no. 11-12, pp. 1441–1446, 2007.
• G. Chartrand, G. L. Johns, K. A. McKeon, and P. Zhang, “On the rainbow connectivity of cages,” in Proceedings of the Thirty-Eighth Southeastern International Conference on Combinatorics, Graph Theory and Computing, vol. 184, pp. 209–222, 2007.
• C. Balbuena, J. Fresan, D. González-Moreno, and M. Olsen, “Rainbow connectivity of Moore cages of girth 6,” Discrete Applied Mathematics, vol. 250, pp. 104–109, 2018. \endinput