Difference labelling of cacti

A graph G is a difference graph iff there exists S ⊂ IN such that G is isomorphic to the graph DG(S) = (V,E), where V = S and E = {{i, j} : i, j ∈ V ∧ |i− j| ∈ V }. It is known that trees, cycles, complete graphs, the complete bipartite graphs Kn,n and Kn,n−1, pyramids and n-sided prisms (n ≥ 4) are difference graphs (cf. [4]). Giving a special labelling algorithm, we prove that cacti with a girth of at least 6 are difference graphs, too.