I need to this for my homework and I've been looking for over an hour but don't really know where to start. A tip would be appreciated!
Assume we have the following graph:
How can you prove that it has a Hamiltonian cycle?
I know there exists a Hamiltonian cycle (drawn below)
A complete (simple) graph is one in which each vertex is connected to each other. It's known that every complete graph on more than two vertices is hamiltonian. Since your graph is complete, it follow from that it has a hamiltonian cycle.
As I recall, there's also a theorem like: If the sum of degrees of every two distinct vertices is at least $n$ in an $n$ vertex graph it has a hamiltonian cycle.