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Let $G$ be a nontrivial connected graph with an edge-coloring$c:E(G)\rightarrow \{1,2,\ldots,q\},$ $q\in \mathbb{N}$, where adjacent edgesmay be colored the same. A tree $T$ in $G$ is called a $rainbow~tree$ if no twoedges of $T$ receive the same color. For a vertex set $S\subseteq V(G)$, a treethat connects $S$ in $G$ is called an {\it $S$-tree}. The minimum number ofcolors that are needed in an edge-coloring of $G$ such that there is a rainbow$S$-tree for every $k$-set $S$ of $V(G)$ is called the {\it $k$-rainbow index}of $G$, denoted by $rx_k(G)$. Notice that an lower bound and an upper bound ofthe $k$-rainbow index of a graph with order $n$ is $k-1$ and $n-1$,respectively. Chartrand et al. got that the $k$-rainbow index of a tree withorder $n$ is $n-1$ and the $k$-rainbow index of a unicyclic graph with order$n$ is $n-1$ or $n-2$. Li and Sun raised the open problem of characterizing thegraphs of order $n$ with $rx_k(G)=n-1$ for $k\geq 3$. In early papers wecharacterized the graphs of order $n$ with 3-rainbow index 2 and $n-1$. In thispaper, we focus on $k=4$, and characterize the graphs of order $n$ with4-rainbow index 3 and $n-1$, respectively.

Graphs with $4$-rainbow index $3$ and $n-1$