Let $G$ be a finite simple graph with vertex set $V(G)$ and edge set $E(G)$.
A function $f : V(G) \rightarrow \mathcal{P}(\{1, 2, \dotsc, k\})$ is a \textit{$k$-rainbow dominating function} on $G$ if
for each vertex $v \in V(G)$ for which $f(v)= \emptyset$, it holds that $\bigcup_{u \in N(v)}f(u) = \{1, 2, \dotsc, k\}$.
The weight of a $k$-rainbow dominating function is the value $\sum_{v \in V(G)}|f(v)|$.
The \textit{$k$-rainbow domination number} $\gamma_{rk}(G)$ is the minimum weight of a $k$-rainbow dominating function on $G$.
In this paper, we initiate the study of $k$-rainbow domination numbers in middle graphs.
We define the concept of a middle $k$-rainbow dominating function, obtain some bounds related to it
and determine the middle $3$-rainbow domination number of some classes of graphs.
We also provide upper and lower bounds for the middle $3$-rainbow domination number of trees in terms of the matching number.
In addition, we determine the $3$-rainbow domatic number for the middle graph of paths and cycles.
Radio number for middle graph of paths
