Upper Boundsand Extreme Results for Conflict-free Vertexconnection Number

Abstract

A path of a vertex-colored graph is conflict-free path, if there exists a color used only on one of its vertices; a vertex-colored graph is conflict-free vertex-connected, if there is a conflict-free path between each pair of distinct vertices of the graph. For a connected graph G, the minimum number of colors required to make G conflict-free vertex-connected is conflict-free vertex- connection number of G, denoted by vcfc(G). In this paper, we first showed an upper bound of vcfc(G) for the general graph by structural method. And then, we gave a partial solution to the conjecture on the conflict-free vertex-connection number by contradiction, posed by Doan and Schiermeyer in [Conflict-free vertex connection number at most 3 and size of graphs, Discus. Math. Graph Theory].