2-rainbow domination in generalized Petersen graphs P(n, 3)

Assume we have a set of k colors and we assign an arbitrary subset of these colors to each vertex of a graph G. If we require that each vertex to which an empty set is assigned has in its neighborhood all k colors, then this assignment is called a k-rainbow dominating function of G. The corresponding invariant γrk(G), which is the minimum sum of numbers of assigned colors over all vertices of G, is called the k-rainbow domination number of G. B. Brešar and T.K. Šumenjak [On the 2-rainbow domination in graphs, Discrete Appl. Math. 155 (2007) 2394–2400] showed that d 4n 5 e ≤ γr2(P(n, k)) ≤ n for any generalized Petersen graph P(n, k), where n and k are relatively prime numbers. And they proposed the question: Is γr2(P(n, 3)) = n for all n ≥ 7 where n is not divisible by 3? In this note, we show that γr2(P(n, 3)) ≤ n − 1 for all n ≥ 13. Moreover, we show that γr2(P(n, 3)) ≤ n − b n 8 c + β , where β = 0 for n ≡ 0, 2, 4, 5, 6, 7, 13, 14, 15 (mod 16) and β = 1 for n ≡ 1, 3, 8, 9, 10, 11, 12 (mod 16). © 2009 Elsevier B.V. All rights reserved.