I have to prove that Connected graph is Eulerian Graph if and only if the number of edges in minimal cut is even.
Implication from left to right. Well if $G$ is Eulerian Graph then degree of each vertex must be even. It's oblivious if i try to disconnect graph by removing all edges connected to the vertex with minimal degree. But minimal cut can be done also by dividing graph into 2 connected graphs (each graph with more then one vertex) and i have no idea how to deal with it.
Implication from right to left - completly no idea.