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;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 rainbow t-colorings with a small number of colors.
An ({r,r+1};g)-cage is a graph with degree set {r,r+1}, girth g, and with the smallest possible order; every such graph is called a semiregular cage. In this article, semiregular cages are shown to be maximally\ud
edge-connected and 2-connected. As a consequence, ({3,4};g)-cages are proved to be maximally connected.Postprint (published version
Let G be an edge-colored graph. A path P of G is said to be rainbow if no two edges of P have the same color. An edge-coloring of G is a rainbow t-coloring if for any two distinct vertices u and v of G there are at least t internally vertex-disjoint rainbow (u, v)-paths. The rainbow t-connectivity rc t (G) of a graph G is the minimum integer j such that there exists a rainbow t-coloring using j colors. A (k; g)-cage is a k-regular graph of girth g and minimum number of vertices denoted n(k; g). In this paper we focus on g = 6. It is known that n(k; 6) ≥ 2(k 2 − k + 1) and when n(k; 6) = 2(k 2 − k + 1) the (k; 6)-cage is called a Moore cage. In this paper we prove that the rainbow k-connectivity of a Moore (k; 6)-cage G satisfies that k ≤ rc k (G) ≤ k 2 − k + 1. It is also proved that the rainbow 3-connectivity of the Heawood graph is 6 or 7.
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