Conflict-free (vertex-)connection numbers of graphs with small diameter

A path in an edge (vertex)-colored graph is called a conflict-free path if there exists at least one color that is used on only one of its edges (vertices). An edge (vertex)-colored graph is called conflict-free (vertex-) connected if for each pair of distinct vertices, there is a conflict-free path connecting them. For a connected graph G, the conflict-free (vertex-) connection number of G, denoted by cfc(G) (or vcfc(G)), is defined as the smallest number of colors that are required to make G conflict-free (vertex-)connected. We use C(G) to denote the subgraph induced by the set of all cut-edges of G. It is easy to see that C(G) is a (possibly empty) forest. Let h(G) = max{cfc(T) : T isacomponentof C(G)}. In this paper, we first give the exact value of cfc(T) for any tree T with diameter 2, 3 or 4. Based on this result, the conflict-free connection number is determined for any graph C with diam(G) <= 4 except for those graphs G with diameter 4 and h(G) = 2. In this case, we give some graphs with conflict-free connection numbers 2 and 3. For the conflictfree vertex-connection number, the exact value of vcfc(G) is determined for any graph G with diam(G) <= 4.