Complexity
The problem of finding a Hamiltonian cycle or path is in FNP; the analogous decision problem is to test whether a Hamiltonian cycle or path exists. The directed and undirected Hamiltonian cycle problems were two of Karp's 21 NP-complete problems. They remain NP-complete even for undirected planar graphs of maximum degree three, for directed planar graphs with indegree and outdegree at most two, for bridgeless undirected planar 3-regular bipartite graphs, and for 3-connected 3-regular bipartite graphs. However, putting all of these conditions together, it remains open whether 3-connected 3-regular bipartite planar graphs must always contain a Hamiltonian cycle, in which case the problem restricted to those graphs could not be NP-complete; see Barnette's conjecture.
In graphs in which all vertices have odd degree, an argument related to the handshaking lemma shows that the number of Hamiltonian cycles through any fixed edge is always even, so if one Hamiltonian cycle is given, then a second one must also exist. However, finding this second cycle does not seem to be an easy computational task. Papadimitriou defined the complexity class PPA to encapsulate problems such as this one.
Read more about this topic: Hamiltonian Path Problem
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