The k -Rainbow Domination Number of C n □ C m

Given a graph G and a set of k colors, assign an arbitrary subset of these colors to each vertex of G . If each vertex to which the empty set is assigned has all k colors in its neighborhood, then the assignment is called a k -rainbow dominating function ( k RDF) of G . The minimum sum of numbers of assigned colors over all vertices of G is called the k -rainbow domination number of graph G , denoted by γ r k ( G ) . In this paper, we focus on the study of the k -rainbow domination number of the Cartesian product of cycles, C n □ C m . For k ≥ 8 , based on the results of J. Amjadi et al. (2017), γ r k ( C n □ C m ) = m n . For ( 4 ≤ k ≤ 7 ) , we give a proof for the new lower bound of γ r 4 ( C n □ C 3 ) . We construct some novel and recursive k RDFs which are good enough and upon these functions we get sharp upper bounds of γ r k ( C n □ C m ) . Therefore, we obtain the following results: (1) γ r 4 ( C n □ C 3 ) = 2 n ; (2) γ r k ( C n □ C m ) = k m n 8 for n ≡ 0 ( mod 4 ) , m ≡ 0 ( mod 4 ) ( 4 ≤ k ≤ 7 ) ; (3) for n ≢ 0 ( mod 4 ) or m ≢ 0 ( mod 4 ) , m n 2 ≤ γ r 4 ( C n □ C m ) ≤ m n 2 + m + n 2 − 1 and k m n 8 ≤ γ r k ( C n □ C m ) ≤ k m n + n 8 + m for 5 ≤ k ≤ 7 . We also discuss Vizing’s conjecture on the k -rainbow domination number of C n □ C m .