k-rainbow domination and some related problems

‎‎Let $G=(V,E)$ be a simple graph and $k\in \mathbb{N}$‎. ‎A $k$-rainbow dominating function of $G$ is a function‎ ‎$f:V\rightarrow 2^{\{1,...,k\}} $ such that for each $v\in V$ with $f(v) =\emptyset$‎, ‎$\bigcup_{u\in N(v)} f(u) = \{1,...,k\}.$ The‎ ‎minimum of $\sum_{v\in V} |f(v)|$ over all $k$-rainbow dominating functions $f$ of $G$ is denoted by‎ ‎$\gamma_{rk}(G)$‎. ‎In this paper‎, ‎we consider small $k$ and study some problems related to $k$-rainbow domination theory‎.