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No (or rather: no, unless P=NP).

If it were so, then there would be a concrete polynomial $p$ that bounded the running time of such an algorithm. Therefore you would be able to detect whether a graph has a Hamiltonian cycle by lying to the algorithm: Claim that the graph has a Hamiltonian cycle, run it for $p(n)$ steps, and then check whether by then it has printed a correct cycle or not.