For an undirected connected graph:
Euler circuit exists if and only if every vertex has even degree.
Euler path exists if and only if exactly or vertices have odd degree.
- If odd vertices: Euler circuit exists (start anywhere)
- If odd vertices: Euler path exists (start at one odd vertex, end at the other)
- If more than odd vertices: no Euler path exists
Why? Each time you enter a vertex, you must leave it. If degree is odd, you'll get stuck. The exceptions are start and end vertices of a path.