(untitled) (European Journal of Pure and Applied Mathematics, 2025)
Let \( G = (V(G), E(G)) \) be a graph. A function \( f \) that assigns to each vertex of G a subset of colors from the set \( {1, 2, …, k} \), i.e., \( f : V(G) → P({1, 2, 3, …, k}) \), is called a hop \( k \)-rainbow dominating function (HkRDF) of \( G \) if for every vertex \( v V(G)\) with f(v)= , we have \( _u N^2_G(v) f(u) = {1, 2, …, k} \) where N^2_G(v) is the set of vertices of G at distance two from v. The weight of \( f \), denoted \( w(f) \), is defined as \( w(f) = _x V(G) |f(x)| \). The hop \( k \)-rainbow domination number of \( G \), denoted \( γ_hrk(G) \), is the minimum weight of a hop \( k \)-rainbow dominating function of \( G \). A hop k-rainbow dominating function of G with weight γ_hrk(G) is a γ_hrk-function of G. In this paper, we initiate the study of hop \( k \)-rainbow domination in graphs. We begin by exploring fundamental properties of this parameter and then establish various bounds on \( γ_hrk(G) \). Furthermore, we identify the graphs for which \( γ_hrk(G) = n \) and determine exact values for certain graph classes, including complete graphs, complete bipartite graphs, paths, and cycles. Additionally, for any positive integer \( a \), we construct connected graphs satisfying \(γ_hr2(G) = γ_r2(G) = a.\) Finally, we provide a characterization of all graphs where \( γ_hr2(G) = n \).