On The Maximum Value Of Conflict-free Vertex-connection Number Of Graphs

A path in a vertex-colored graph is called conflict-free if there is a color used on ex-actly 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.The conflict-free vertex-connection number,denoted by vcf c(G),is defined as the smallest number of col-ors required to make G conflict-free vertex-connected.Li et al.[12]conjectured that for a connected graph G of order n,vcf c(G)?vcf c(P_n).We confirm that conjecture is true and pose two relevant conjectures.