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

Abstract

A path in a vertex-coloured graph is called conflict-free if there is a colour used on exactly one of its vertices. A vertex-coloured graph is said to be conflict-free vertex-connected if any two distinct vertices of the graph are connected by a conflict-free vertex-path. The conflict-free vertex-connection number, denoted by vcfc(G), is the smallest number of colours needed in order to make G conflict-free vertex-connected. Clearly, vcfc(G) ≥ 2 for every connected graph on n ≥ 2 vertices. Our main result of this paper is the following. Let G be a connected graph of order n. If |E(G)|≥(n-6 2)+7, then vcfc(G) ≤ 3. We also show that vcfc(G) ≤ k + 3-t for every connected graph G with k cut-vertices and t being the maximum number of cut-vertices belonging to a block of G.