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Recently, I studied how to prove a problem is NP-complete. From the book Introduction to Algorithms (CLRS, 3rd edition), there is an example of reducing vertex cover to Hamiltonian cycle. I believe that it is possible to directly reduce clique to Hamiltonian cycle in an undirected graph, but how to convert it in polynomial time?

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Hongyu Shen
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You can reduce clique to SAT (Cook's theorem). You can reduce SAT to Hamiltonian cycle (see e.g. Easy reduction from 3SAT to Hamiltonian path problem for the main ideas). Compose these two, and you get a reduction from clique to Hamiltonian cycle.

D.W.
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