  Let $G$ be a nontrivial connected graph with an edge-coloring $c:
E(G)\rightarrow \{1,2,...,q\},$ $q \in \mathbb{N}$, where adjacent edges may be
colored the same. A tree $T$ in $G$ is a $rainbow tree$ if no two edges of $T$
receive the same color. For a vertex subset $S\subseteq V(G)$, a tree that
connects $S$ in $G$ is called an $S$-tree. The minimum number of colors that
are needed in an edge-coloring of $G$ such that there is a rainbow $S$-tree for
each $k$-subset $S$ of $V(G)$ is called $k$-rainbow index, denoted by
$rx_k(G)$. In this paper, we first determine the graphs whose 3-rainbow index
equals 2, $m,$ $m-1$, $m-2$, respectively. We also obtain the exact values of
$rx_3(G)$ for regular complete bipartite and multipartite graphs and wheel
graphs. Finally, we give a sharp upper bound for $rx_3(G)$ of 2-connected
graphs and 2-edge connected graphs, and graphs whose $rx_3(G)$ attains the
upper bound are characterized.

Combinatorics