Abstract
Arainbowt-coloringof at-connected graphGis an edge coloring such that for any two distinct verticesuandvofGthere are at leasttinternally vertex-disjoint rainbow(u,v)-paths. In this work, we apply a Rank Genetic Algorithm to search for rainbowt-colorings of the family of Moore cages with girth six(t;6)-cages. We found that an upper bound in the number of colors needed to produce a rainbow 4-coloring of a(4;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 rainbowt-colorings with a small number of colors.
Highlights
Introduction and DefinitionsEvolutionary algorithms have been applied to a wide variety of engineering problems [1], and they have been applied to mathematics problems
Jong and Spears [2] showed that Genetic Algorithms (GA) can be used to solve NP-Complete problems, [3] applied a GA to a geometry problem, and [4] solved nonlinear algebraic equations by using a GA
With the Rank Genetic Algorithm (Rank GA) we could see that it is unlikely that the upper bound for (3; 6)-cages is less than 7, since with 6 colors no solution was found after having let the algorithm run for a long time
Summary
Introduction and DefinitionsEvolutionary algorithms have been applied to a wide variety of engineering problems [1], and they have been applied to mathematics problems. Jong and Spears [2] showed that Genetic Algorithms (GA) can be used to solve NP-Complete problems, [3] applied a GA to a geometry problem, and [4] solved nonlinear algebraic equations by using a GA. We have successfully applied a Rank Genetic Algorithm (Rank GA) [5] to the graph theory problem of finding the rainbow connection number of a graph (rc(G)). Chakraborty et al [6] proved that, for a given graph G, deciding whether rc(G) = 2 is NP-Complete and that it is NP-Complete to decide whether a given edge-colored (with an unbounded number of colors) graph is rainbow connected. This problem has not been approached in this way nor using any other heuristic algorithms
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