On the Complexity of Determining Whether there is a Unique Hamiltonian Cycle or Path

The decision problems of the existence of a Hamiltonian cycle or of a Hamiltonian path in a given directed or undirected graph, and of the existence of a truth assignment satisfying a given Boolean formula C, are well-known NP-complete problems. Here we study the problems of the uniqueness of a Hamiltonian cycle or path in an undirected, directed or oriented graph, and show that they have the same complexity, up to polynomials, as the problem U-SAT of the uniqueness of an assignment satisfying C. As a consequence, these Hamiltonian problems are NP-hard and belong to the class DP, like U-SAT.