2-rainbow domination in generalized Petersen graphs P(n,3) P ( n , 3 )
Assume we have a set of k k colors and we assign an arbitrary subset of these colors to each vertex of a graph G G . If we require that each vertex to which an empty set is assigned has in its neighborhood all k k colors, then this assignment is called a k k -rainbow dominating function of G G . The corresponding invariant ?rk(G) ? r k ( G ) , which is the minimum sum of numbers of assigned colors over all vertices of G G , is called the k k -rainbow domination number of G G . B. Brear and T.K. umenjak [On the 2-rainbow domination in graphs, Discrete Appl. Math. 155 (2007) 23942400] showed that ? 4 n 5 ? ? ? r 2 ( P ( n , k ) ) ? n for any generalized Petersen graph P(n,k) P ( n , k ) , where n n and k k are relatively prime numbers. And they proposed the question: Is ?r2(P(n,3))=n ? r 2 ( P ( n , 3 ) ) = n for all n?7 n ? 7 where n n is not divisible by 3? In this note, we show that ?r2(P(n,3))?n?1 ? r 2 ( P ( n , 3 ) ) ? n ? 1 for all n?13 n ? 13 . Moreover, we show that ? r 2 ( P ( n , 3 ) ) ? n ? ? n 8 ? + ? , where ?=0 ? = 0 for n?0,2,4,5,6,7,13,14,15 n ? 0 , 2 , 4 , 5 , 6 , 7 , 13 , 14 , 15 (mod 16) and ?=1 ? = 1 for n?1,3,8,9,10,11,12 n ? 1 , 3 , 8 , 9 , 10 , 11 , 12 (mod 16).
2-rainbow domination domination the generalized petersen graph