  The decision problems of the existence of a Hamiltonian cycle or of a
Hamiltonian path in a given graph, and of the existence of a truth assignment
satisfying a given Boolean formula $C$, are well-known {\it NP}-complete
problems. Here we study the problems of the {\it 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 {\it NP}-hard and belong to the class~{\it DP}, like U-SAT.
