Conflict-Free Vertex-Connections of Graphs

A path in a vertex-colored graph is called if there is a color used on exactly one of its vertices. A vertex-colored graph is said to be if any two vertices of the graph are connected by a conflict-free path. This paper investigates the question: for a connected graph , what is the smallest number of colors needed in a vertex-coloring of in order to make conflict-free vertex-connected. As a result, we get that the answer is easy for 2-connected graphs, and very difficult for connected graphs with more cut-vertices, including trees.