Conflict-Free Vertex Connection Number At Most 3 and Size of Graphs

A path in a vertex-coloured graph is called if there is a colour used on exactly one of its vertices. A vertex-coloured graph is said to be if any two distinct vertices of the graph are connected by a conflict-free vertex-path. The , denoted by ), is the smallest number of colours needed in order to make conflict-free vertex-connected. Clearly, ) ≥ 2 for every connected graph on ≥ 2 vertices. Our main result of this paper is the following. Let be a connected graph of order . If , then ) ≤ 3. We also show that ) ≤ + 3 − for every connected graph with cut-vertices and being the maximum number of cut-vertices belonging to a block of