Conflict-Free Vertex-Connections of Graphs

• Main Authors: Li Xueliang, Zhang Yingying, Zhu Xiaoyu, Mao Yaping, Zhao Haixing, Jendrol’ Stanislav
• Format: Article
• Language: English
• Published: Sciendo 2020-02-01
• Series: Discussiones Mathematicae Graph Theory
• Subjects: vertex-coloring; conflict-free vertex-connection; 2-connected graph; tree; 05c15; 05c40; 05c75
• Online Access: https://doi.org/10.7151/dmgt.2116

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.