A few doubts regarding graph theory:
When considering cycles in a connected graph, we have cycles that go both clockwise and anti-clockwise. Do we consider these as two separate cycles?
In a complete graph such as $K_3$, how many Hamiltonian cycles exist? My guess is that it is $3!$ because each vertex connects to every other vertex and any permutation of the vertices gives you a cycle. But when I look it up online, it says there are only 2. How and why?
Consider simple cycles, not Hamiltonian or Eulerian. Am I right if I say that $K_3$ has $3!$ simple cycles?