  For a positive integer $k$, a $k$-rainbow dominating function ($k$RDF) on a
digraph $D$ is a function $f$ from the vertex set $V(D)$ to the set of all
subsets of $\{1,2,\ldots,k\}$ such that for any vertex $v$ with
$f(v)=\emptyset$, $\bigcup_{u\in N^-(v)}f(u)=\{1,2,\ldots,k\}$, where $N^-(v)$
is the set of in-neighbors of $v$. The weight of a $k$RDF $f$ is defined as
$\sum_{v\in V(D)}|f(v)|$. A $k$RDF $f$ on $D$ with no isolated vertex is called
a total $k$-rainbow dominating function if the subdigraph of $D$ induced by the
set $\{v\in V(D):f(v)\ne\emptyset\}$ has no isolated vertex. The total
$k$-rainbow domination number is the minimum weight of a total $k$-rainbow
dominating function on $D$. In this paper, we establish some bounds for the
total $k$-rainbow domination number and we give the total $k$-rainbow
domination number of some digraphs.

Combinatorics