Geodesic
Conflict-Free Vertex-Connections of Graphs
Discussiones Mathematicae. Graph Theory, Tome 40 (2020) no. 1, pp. 51-65.

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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.
Keywords: vertex-coloring, conflict-free vertex-connection, 2-connected graph, tree 
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Li, Xueliang; Zhang, Yingying; Zhu, Xiaoyu; Mao, Yaping; Zhao, Haixing; Jendrol’, Stanislav. Conflict-Free Vertex-Connections of Graphs. Discussiones Mathematicae. Graph Theory, Tome 40 (2020) no. 1, pp. 51-65. http://geodesic.mathdoc.fr/item/DMGT_2020_40_1_a3/

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