Some Results on the 3-Vertex-Rainbow Index of a Graph

Let G be a nontrivial connected graph with a vertex-coloring c : V ( G ) → { 1 , 2 , … , q } , q ∈ N . For a set S ⊆ V ( G ) and | S | ≥ 2 , a subtree T of G satisfying S ⊆ 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 ) \ S have distinct colors. Let k be a fixed integer with 2 ≤ k ≤ | 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 r v x 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.