In this problem, we consider the edge connectivity of a simple undirected graph, which is the minimum number of edges one can remove to disconnect it.
Prove that if G is a connected simple undirected graph where every vertex's degree is a multiple of 2, then one must remove at least 2 edges in order to disconnect the graph. (It should be noted that removing a vertex does not necessarily disconnect a simple graph.)
Does the statement above remain true if the number 2 is replaced with any positive integer k? If so, prove it. If not, give a counterexample.
Typically I would write where I am for a problem like this, but I have no idea how to approach this proof. Any and all help would be much appreciated.