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A Hamiltonian graph (directed or undirected) is a graph that contains a Hamiltonian cycle, that is, a cycle that visits every vertex exactly once.

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Which are good algorithms for finding Hamiltonian path (not necessarily a circle) up to now?

The most efficient exact algorithm for finding Hamilton paths and circuits is "A Search Procedure for Hamilton Paths and Circuits." J. ACM 21 (Oct. 1974), pp 576-580; DOI: 10.1145/321850.321854.
Frank Rubin's user avatar
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Efficient Hamiltonian cycle algorithms for graph classes

Even though the Hamilton path problem is NP-hard, in practice it is usually easy. An efficient algorithm has been known for 50 years. It is "A Search Procedure for Hamilton Paths and Circuits." J. A …
Frank Rubin's user avatar