Necessary and sufficient conditions for Euler paths
Theorem 2. A connected multigraph has an Euler path but not an
Euler circuit if and only if it has exactly two vertices of odd degree.
(ONLY IF) Assume the graph has an Euler path but not a circuit.
Notice that every time the path passes through a vertex, it contributes
2 to the degree of the vertex (1 when it enters, 1 when it leaves).
Obviously the first and the last vertices will have odd degree and all
the other vertices - even degree.
(IF) Assume exactly two vertices, u and v, have odd degree.
If we connect these two vertices, then every vertex will have even degree.
By Theorem 1, there is an Euler circuit in such a graph.
If we remove the added edge {u,v} from this circuit, we will get an
Euler path for the original graph.