A problem I'm working on reduced to finding a closed loop that visited every node of a cube. I remembered this was called a Hamiltonian cycle, and so I needed to find the number of Hamiltonian cycles.
I looked up there are 12 distinct directed Hamiltonian cycles for the cube (via MathWorld)
I worked out the 6 undirected cycles (as each path can go in 2 directions, this makes for a total of 6x2 = 12 directed cycles).
Is there a way to prove these are all of the distinct undirected Hamiltonian cycles? Or is it found by method of exhaustion (i.e., trial and error and basically finding that no other path works).
I'd be happy to cite Mathworld saying this is the answer, but it feels somewhat unsatisfying not to give further reason.
I did many, many web searches but kept finding complicated course notes unrelated to this question. I suspect it would be a basic graph theory question, but somehow a reference eludes me.
Please let me know if there's a way to prove there are 12 directed Hamiltonian cycles in a cubical graph. References would be much appreciated too--I am happy to work through some graph theory to understand why. Thanks!