The 3-Vertex-Rainbow Index of 2-(Edge) Connected Graphs

Let [Formula: see text] be a vertex-colored graph. For a vertex set [Formula: see text] of at least two vertices, a tree [Formula: see text] that connects [Formula: see text] in [Formula: see text] is vertex-rainbow if no two vertices of [Formula: see text] have the same color, such a tree is called a vertex-rainbow [Formula: see text]-tree or a vertex-rainbow tree connecting [Formula: see text]. Let [Formula: see text] be a fixed integer with [Formula: see text], [Formula: see text] is said to be vertex-rainbow [Formula: see text]-tree connected if every [Formula: see text]-subset [Formula: see text] of [Formula: see text] has a vertex-rainbow [Formula: see text]-tree. The [Formula: see text]-vertex-rainbow index [Formula: see text] of a graph [Formula: see text] is the minimum number of colors are needed in order to make [Formula: see text] vertex-rainbow [Formula: see text]-tree connected. In this paper, we focus on [Formula: see text]. When [Formula: see text] is [Formula: see text]-connected or [Formula: see text]-edge-connected, we provide a sharp upper bound for [Formula: see text], respectively, and determine the graphs [Formula: see text], where [Formula: see text] reaches the upper bound.