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.