Multiple of Hamiltonian Cycles I'm currently confused whether a graph should contain strictly one distinct Hamiltonian Cycle. (given that [1,2,3,4,1] and [2,3,4,1,2] are the same).

I was wondering if, by definition, there can be multiple distinct Hamiltonian Cycles in a graph? something like H1 = [1,2,3,1] and H2 = [6,7,8,6]

can someone kindly confirm?

(addt'l sources and references would really be great so I can dive in deeper)

Cheers!
Recall a Hamiltonian cycle visits each vertex of a graph exactly once. Thus, the cycles H1 and H2 in your second example can't be Hamiltonian cycles in the same graph.

In your first example, the two cycles could be distinct Hamiltonian cycles in e.g. the complete graph on 4 vertices. That is to say, there can definitely be multiple Hamiltonian cycles in a graph.

If you care about naming conventions, you can call two cycles that don't share any vertices vertex-disjoint cycles. Similarly, two cycles can be edge-disjoint if they don't share edges.