Let G be a nontrivial connected graph with a vertex-coloring c: \(V(G)\rightarrow \{1,2,\ldots ,q\},q\in N\). For a set \(S\subseteq V(G)\) and \(|S|\ge 2\), a subtree T of G satisfying \(S\subseteq V(T)\) is said to be an S-Steiner tree or simply S-tree. The S-tree T is called a vertex-rainbow S-tree if the vertices of \(V(T)\setminus S\) have distinct colors. Let k be a fixed integer with \(2\le k\le |V(G)|\), if every k-subset S of V(G) has a vertex-rainbow S-tree, then G is said to be vertex-rainbow k-tree connected. The k-vertex-rainbow index of G, denoted by \(rvx_{k}(G)\), is the minimum number of colors that are needed in order to make G vertex-rainbow k-tree connected. In this paper, we study the 3-vertex-rainbow index of unicyclic graphs and complementary graphs, respectively.

令G为顶点着色为c的非平凡连通图：\（V（G）\ rightarrow \ {1,2，\ ldots，q \}，q \ in N \）。对于一组\（S \ subseteq V（G）\）和\（| S | \ GE 2 \），子树Ť的ģ满足\（S \ subseteq V（T）\）被说成是一个小号- Steiner树或简单的S-树。如果\（V（T）\ setminus S \）的顶点具有不同的颜色，则S树T被称为顶点彩虹S树。设k为\（2 \ le k \ le | V（G）| \）的固定整数，如果每个ķ -subset小号的V（ģ）具有顶点彩虹小号-tree，然后ģ据说是顶点彩虹ķ -tree连接。G的k-顶点-彩虹索引由\（rvx_ {k}（G）\）表示，是使G顶点-彩虹k-树连接所需的最小颜色数。在本文中，我们分别研究了单圈图和互补图的3-顶点-彩虹指数。