I'm preparing for a test next week about graph theory. One of the example exercises is about proving that a closed knight's tour exists on an 8-by-8 chess board by only using basic theorems such as Ore's theorem or some other not too complex way. I know that it is possible to construct a knights tour, but I think that that's the right way to proof this theorem as we only have half an hour to solve the question and finding a knights tour by hand (no computers) isn't that simple.
This problem is equivalent to finding a Hamiltonian cycle in a graph representing all of the knight's legal moves between all the squares.