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

• Authors: Guangjun Xu
• Venue: Discrete Applied Mathematics
• Year: 2009

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 @c"r"k(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. Bresar and T.K. Sumenjak [On the 2-rainbow domination in graphs, Discrete Appl. Math. 155 (2007) 2394-2400] showed that @?4n5@?@?@c"r"2(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 @c"r"2(P(n,3))=n for all n=7 where n is not divisible by 3? In this note, we show that @c"r"2(P(n,3))@?n-1 for all n=13. Moreover, we show that @c"r"2(P(n,3))@?n-@?n8@?+@b, where @b=0 for n=0,2,4,5,6,7,13,14,15 (mod 16) and @b=1 for n=1,3,8,9,10,11,12 (mod 16).