# 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!

• In your second example, H1 and H2 can't be Hamiltonian cycles in the same graph since a Hamiltonian cycle includes every vertex exactly once. In your first example, the two cycles could be distinct Hamiltonian cycles in a complete graph on 4 vertices. – Juho May 19 '15 at 15:54
• I see.. so for the second example, H1 and H2 would simply be called cycles right? is there a distinct name for cycles that doesn't contain same vertices? – Kevin Lloyd Bernal May 19 '15 at 16:00
• If you want to, you can call them vertex-disjoint cycles (i.e. they don't share any vertices). – Juho May 19 '15 at 16:17