Rainbow connectivity of Moore cages of girth 6

Abstract

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.

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

“…Chartrand et al [20] showed that the rainbow 3connectivity of the (3; 6)-cage (the Moore cage known as Heawood graph) is between 5 and 7 inclusive. Recently, Balbuena et al [21] proved that it is not 5, and they bounded the rainbow -connectivity of ( ; 6)-cages as follows: if is a Moore ( ; 6)-cage, then…”

“…If is a ( , 6)-cage, then for = 3, = 4, and = 5, respectively, we have 3 ( ) ≤ 7, 4 ( ) ≤ 13, and 5 ( ) ≤ 21, respectively. In this paper we use a genetic algorithm to search for rainbow -colorings of ( ; 6)-cages with = 3 and = 4 [21] in order to see if the upper bound for ( ) can be improved.…”