I know that the Hamiltonian cycle problem in a directed/undirected graph is NP complete. The proofs I know rely on a reduction from 3-SAT. However, in these reductions the constructed graph is not simple (i.e., some pairs of nodes have multiple edges between them albeit with different orientations). I suspect that the problem remains NP-complete even if the attention is restricted to simple directed graphs (where there is at most one edge in between a pair of nodes). However, I wasn't able to find a reference.
Question: Is the Hamiltonian cycle problem still NP-complete when we focus on simple directed graphs?