Question

A Euler circuit touches every __________ while a Hamiltonian circuit touches every __________.

Answer 1

Hint: If those are all 4 vertices of a graph, the edges are invisible, and the line I drew is the path, it would be called a Hamiltonian Path.

Answer 2

then what does a euler circuit touch?

Answer 3

So the hamiltonian path touched the edges. The other possibility is the edges.