Is it true that an Euler path should have two vertices of odd degree and an Euler circuit should have no vertices of odd degree? Is it therefore impossible to have a graph with both an Euler path and an Euler circuit?
This link (which you have linked in the comment to the question) states that having Euler path and circuit are mutually exclusive. The definition of Euler path in the link is, however, wrong - the definition of Euler path is that it's a trail, not a path, which visits every edge exactly once. And in the definition of trail, we allow the vertices to repeat, so, in fact, every Euler circuit is also an Euler path.