  A {\it 2-rainbow domination function} of a graph $G$ is a function $f$ that
assigns to each vertex a set of colors chosen from the set $\{1,2\}$, such that
for any $v\in V(G)$, $f(v)=\emptyset$ implies $\bigcup_{u\in
N(v)}f(u)=\{1,2\}$. The {\it 2-rainbow domination number $\gamma_{r2}(G)$} of a
graph $G$ is the minimum $w(f)=\Sigma_{v\in V}|f(v)|$ over all such functions
$f$. Let $G$ be a connected graph of order $|V(G)|=n\geq 3$. We prove that
$\gamma_{r2}(G)\leq 3n/4$ and we characterize the graphs achieving equality. We
also prove a lower bound for 2-rainbow domination number of a tree using its
domination number. Some other lower and upper bounds of $\gamma_{r2}(G)$ in
terms of diameter are also given.

Combinatorics