  A tree in an edge-colored graph is said to be rainbow if no two edges on the
tree share the same color. An edge-coloring of $G$ is called 3-rainbow if for
any three vertices in $G$, there exists a rainbow tree connecting them. The
3-rainbow index $rx_3(G)$ of $G$ is defined as the minimum number of colors
that are needed in a 3-rainbow coloring of $G$. This concept, introduced by
Chartrand et al., can be viewed as a generalization of the rainbow connection.
In this paper, we study the 3-rainbow index by using connected three-way
dominating sets and 3-dominating sets. We shown that for every connected graph
$G$ on $n$ vertices with minimum degree at least $\delta$ ($3\leq\delta\leq5$),
$rx_{3}(G)\leq \frac{3n}{\delta+1}+4$, and the bound is tight up to an additive
constant; whereas for every connected graph $G$ on $n$ vertices with minimum
degree at least $\delta$ ($\delta\geq3$), we get that $rx_{3}(G)\leq
n\frac{ln(\delta+1)}{\delta+1}(1+o_{\delta}(1))+5$. In addition, we obtain some
tight upper bounds of the 3-rainbow index for some special graph classes,
including threshold graphs, chain graphs and interval graphs.
