From How to think in graphs: An illustrative introduction to Graph Theory and its applications by Vardan Grigoryan (vardanator)

**Theorem**. A finite undirected connected graph is an Euler graph if and only if **exactly two** vertices are of **odd degree** or **all vertices** are of **even degree**. In the latter case, every Euler path of the graph is a circuit, and in the former case, none is. [1]