An FPT algorithm for Hamiltonian cycle running parameterized by treewidth
If you want an FPT algorithm for the problem (parameterized by treewidth $t$), you want an algorithm working in time $f(t) \cdot n^{O(1)}$, where $f$ is any computable function (depending solely on $t$). Of course, it would be nice to make $f$ as appealing as possible.

In addition to the mentioned algorithm running in $O(t^t n)$ time, you can also get a faster (randomized) algorithm using the Cut'n'Count technique of Cygan et al. [1]. In particular, you get an algorithm running in time $4^t n^{O(1)}$. It is also possible to get a deterministic algorithm working in $c^t n^{O(1)}$ (for some small constant $c$) using a rank-based approach of [2].



[1] Cygan, M., Nederlof, J., Pilipczuk, M., Pilipczuk, M., van Rooij, J. M., &amp; Wojtaszczyk, J. O. "Solving connectivity problems parameterized by treewidth in single exponential time." IEEE 52nd Annual Symposium on Foundations of Computer Science (FOCS), 2011.

[2] Bodlaender, H. L., Cygan, M., Kratsch, S., &amp; Nederlof, J. "Deterministic single exponential time algorithms for connectivity problems parameterized by treewidth." Automata, Languages, and Programming. Springer Berlin Heidelberg, 2013. 196-207.