Let G be a nontrivial connected graph with an edge-coloring
c:E(G)β{1,2,β¦,q},qβN, where adjacent edges
may be colored the same. A tree T in G is a rainbowtree if no two edges
of T receive the same color. For a vertex set SβV(G), the tree
connecting 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-set S of V(G) is called the k-rainbow index of G, denoted by
rxkβ(G). In \cite{Zhang}, they got that the k-rainbow index of a tree is
nβ1 and the k-rainbow index of a unicyclic graph is nβ1 or nβ2. So
there is an intriguing problem: Characterize graphs with the k-rainbow index
nβ1 and nβ2. In this paper, we focus on k=3, and characterize the graphs
whose 3-rainbow index is nβ1 and nβ2, respectively.Comment: 14 page