Hamiltonian cycles in (2,3,c)(2,3,c)-circulant digraphs

Let DD be the circulant digraph with nn vertices and connection set {2,3,c}{2,3,c}. (Assume DD is loopless and has outdegree 3.) Work of S. C. Locke and D. Witte implies that if nn is a multiple of 6, c∈{(n/2)+2,(n/2)+3}c∈{(n/2)+2,(n/2)+3}, and cc is even, then DD does not   have a hamiltonian cycle. For all other cases, we construct a hamiltonian cycle in DD.