§ Remembering Eulerian and Hamiltonian cycles

I used to keep forgetting the difference. Here's how I remember it now. We know that an euler tour always exists for a tree. Indeed, it's a handy data structure that can be used to convet LCA (lowest common ancestor) into RMQ(range minimum query). So, the "Euler tour" must exist for a tree. See that when we perform a tour on the tree, we definitely walk a vertex twice (once when entering, once when exiting). It seems like we walk the (undirected) edges twice as well. However, if we consider the edges as directed edges , then we're only walking the edges once.

• So an euler tour must correspond to a tour where we walk over each edge exactly once.
• A hamiltonian tour must (by complementarity) correspond to a tour where we over each vertex exactly once.