Let G be a nontrivial connected graph with an edge-coloring c:E(G)β{1,2,...,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 subset SβV(G), a tree that
connects 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-subset S of V(G) is called k-rainbow index, denoted by
rxkβ(G). In this paper, we first determine the graphs whose 3-rainbow index
equals 2, m,mβ1, mβ2, respectively. We also obtain the exact values of
rx3β(G) for regular complete bipartite and multipartite graphs and wheel
graphs. Finally, we give a sharp upper bound for rx3β(G) of 2-connected
graphs and 2-edge connected graphs, and graphs whose rx3β(G) attains the
upper bound are characterized.Comment: 13 page