Conflict-free vertex-connections of graphs
Stanislav Jendrol′, Xueliang Li, Yaping Mao, Yingying Zhang, Haixing Zhao, Xiaoyu Zhu · Discussiones Mathematicae Graph Theory · 2018
A path in a vertex-colored graph is called conflict-free if there is a color used on exactly one of its vertices. A vertex-colored graph is said to be conflict-free vertex-connected if any two vertices of the graph are connected by a conflict-free path. This paper investigates the question: for a connected graph G, what is the smallest number of colors needed in a vertex-coloring of G in order to make G 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.