On k-rainbow domination in regular graphs

The $k$-rainbow domination problem is studied for regular graphs. We prove that the $k$-rainbow domination number ${\gamma }_{rk}(G)$ of a $d$-regular graph for $d\le k\le 2d$ is bounded below by $\displaystyle{\left\lceil kn/2d\right\rceil}$,where$n$ is the order of a graph. We determine necessary conditions for regular graphs to attain this bound and find several examples. As an application, we determine exact $k$-rainbow domination numbers for all cubic Cayley graphs over abelian groups.