2-Rainbow Domination of the Circulant Graph C(n;{1,3})

Let G be a graph where for each vertex, a subset of a set of k colors is assigned. If for each vertex to which an empty set is assigned, its neighborhood contains all k colors, then such an assignment is called a k-rainbow dominating function of G. The corresponding invariant. gamma(rk) (G), which is the minimum sum of the cardinalities of the subsets assigned by a k-rainbow dominating function of G, is called the k-rainbow domination number of G. In this paper, we study the 2-rainbow domination number of the Circulant graph C(n;{1,3}), and we show that gamma(rk) (C(n;{1,3}) = 2right perpendicular n/5 left perpendicular + alpha, where alpha = 0 for n equivalent to 0 (mod 5), alpha = 0 for n equivalent to 1,2 (mod 5) and alpha = 2 for n equivalent to 3,4 (mod 5).