  The $k$-rainbow index $rx_k(G)$ of a connected graph $G$ was introduced by
Chartrand, Okamoto and Zhang in 2010. As a natural counterpart of the
$k$-rainbow index, we introduced the concept of $k$-vertex-rainbow index
$rvx_k(G)$ in this paper. For a graph $G=(V,E)$ and a set $S\subseteq V$ of at
least two vertices, \emph{an $S$-Steiner tree} or \emph{a Steiner tree
connecting $S$} (or simply, \emph{an $S$-tree}) is a such subgraph $T=(V',E')$
of $G$ that is a tree with $S\subseteq V'$. For $S\subseteq V(G)$ and $|S|\geq
2$, an $S$-Steiner tree $T$ is said to be a \emph{vertex-rainbow $S$-tree} if
the vertices of $V(T)\setminus S$ have distinct colors. For a fixed integer $k$
with $2\leq k\leq n$, the vertex-coloring $c$ of $G$ is called a
\emph{$k$-vertex-rainbow coloring} if for every $k$-subset $S$ of $V(G)$ there
exists a vertex-rainbow $S$-tree. In this case, $G$ is called
\emph{vertex-rainbow $k$-tree-connected}. The minimum number of colors that are
needed in a $k$-vertex-rainbow coloring of $G$ is called the
\emph{$k$-vertex-rainbow index} of $G$, denoted by $rvx_k(G)$. When $k=2$,
$rvx_2(G)$ is nothing new but the vertex-rainbow connection number $rvc(G)$ of
$G$. In this paper, sharp upper and lower bounds of $srvx_k(G)$ are given for a
connected graph $G$ of order $n$,\ that is, $0\leq srvx_k(G)\leq n-2$. We
obtain the Nordhaus-Guddum results for $3$-vertex-rainbow index, and show that
$rvx_3(G)+rvx_3(\overline{G})=4$ for $n=4$ and $2\leq
rvx_3(G)+rvx_3(\overline{G})\leq n-1$ for $n\geq 5$. Let $t(n,k,\ell)$ denote
the minimal size of a connected graph $G$ of order $n$ with $rvx_k(G)\leq
\ell$, where $2\leq \ell\leq n-2$ and $2\leq k\leq n$. The upper and lower
bounds for $t(n,k,\ell)$ are also obtained.
