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 $d\leq k\leq 2d$ 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.

中文翻译：

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关于正则图中的 k-rainbow 支配

$k$-rainbow 支配问题是针对正则图研究的。我们证明 $d\leq k\leq 2d$ 的 $d$-正则图的 $k$-rainbow 支配数 $\gamma_{rk}(G)$ 的下界为 $\displaystyle{\left\ lceil kn/2d\right\rceil}$，其中 $n$ 是图的阶数。我们确定了正则图达到这个界限的必要条件，并找到了几个例子。作为一个应用，我们确定了阿贝尔群上所有三次凯莱图的精确 $k$-rainbow 支配数。