  A tree in an edge-colored graph $G$ is said to be a rainbow tree if no two
edges on the tree share the same color. Given two positive integers $k$, $\ell$
with $k\geq 3$, the \emph{$(k,\ell)$-rainbow index} $rx_{k,\ell}(G)$ of $G$ is
the minimum number of colors needed in an edge-coloring of $G$ such that for
any set $S$ of $k$ vertices of $G$, there exist $\ell$ internally disjoint
rainbow trees connecting $S$. This concept was introduced by Chartrand et al.,
and there have been very few results about it. In this paper, we investigate
the $(k,\ell)$-rainbow index for complete bipartite graphs and complete
multipartite graphs. Some asymptotic values of their $(k,\ell)$-rainbow index
are obtained.
