This graph is an Hamiltionian, but NOT Eulerian. Since the degree of $v_{i_2}$ is 2, we can walk to a vertex $v_{i_3}\neq v_{i_2}$ and continue this process. The above graph is an Euler graph as a 1 b 2 c 3 d 4 e 5 c 6 f 7 g covers all the edges of the graph. Proof: If G is Eulerian then there is an Euler circuit, P, in G. Every time a vertex is listed, that accounts for two edges adjacent to that vertex, the one before it in the list and the one after it in the list. Jaeger used them to prove his 4-Flow Theorem [4, Proposition 10]). Review MR#6557 Our approach to Theorem1.1is to reduce it to the following special case: Proposition 1.3. : The claim holds for all graphs with $|E|