Abstract :
Assume we have a set of image colors and we assign an arbitrary subset of these colors to each vertex of a graph image. If we require that each vertex to which an empty set is assigned has in its neighborhood all image colors, then this assignment is called a image-rainbow dominating function of image. The corresponding invariant image, which is the minimum sum of numbers of assigned colors over all vertices of image, is called the image-rainbow domination number of image. B. Brešar and T.K. Šumenjak [On the 2-rainbow domination in graphs, Discrete Appl. Math. 155 (2007) 2394–2400] showed that image for any generalized Petersen graph image, where image and image are relatively prime numbers. And they proposed the question: Is image for all image where image is not divisible by 3? In this note, we show that image for all image. Moreover, we show that image, where image for image (mod 16) and image for image (mod 16).
