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

Abstract

A path in an(a) edge(vertex)-colored graph is called a conflictfree path if there exists a color used on only one of its edges(vertices). An(A) 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 cf c(G)(or vcf c(G)), is defined as the smallest number of colors that are required to make G conflictfree (vertex-)connected. In this paper, we first give the exact value cf c(T ) for any tree T with diameters 2, 3 and 4. Based on this result, the conflict-free connection number is determined for any graph G 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 number 2 and 3, respectively. For the conflict-free vertexconnection number, the exact value vcf c(G) is determined for any graph G with diam(G) ≤ 4.