On graceful difference labelings of disjoint unions of circuits

A graceful difference labeling (gdl for short) of a directed graph G with vertex set V is a bijection f:V→{1,…,|V|} such that, when each arc $uv$ is assigned the difference label f(v)−f(u), the resulting arc labels are distinct. We conjecture that all disjoint unions of circuits have a gdl, except in two particular cases. We prove partial results which support this conjecture.