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

Abstract: 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 edges may be colored the same. A tree $T$ in $G$ is called a $rainbow~tree$ if no two edges of $T$ receive the same color. For a vertex set $S\subseteq V(G)$, a tree that connects $S$ in $G$ is called an {\it $S$-tree}. The minimum number of colors that are needed in an edge-coloring of $G$ such that there is a rainbow $S$-tree for 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 of the $k$-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 with order $n$ 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 the graphs of order $n$ with $rx_k(G)=n-1$ for $k\geq 3$. In early papers we characterized the graphs of order $n$ with 3-rainbow index 2 and $n-1$. In this paper, we focus on $k=4$, and characterize the graphs of order $n$ with 4-rainbow index 3 and $n-1$, respectively.