I have a doubt in the only if part of the proof i.e. if a graph G is connected and it's all vertices has even degree then the graph is eulerian. In this we prove two claims- In the longest path X=(v0,e1,v1,e2,.....,em,vm)
v0=vm (v0 is initial vertex, vm is the last vertex where m is the number of vertices)
set of all ei's in X is the set of edges in G[E(G)]
Doubt- in claim 2 we assume that let the set of edges ei's in X be not equal to E(G) and then end up in a contradiction but how? We assume an edge not in E(X) and say there is a longest path with that edge?
Question - Proof every graph is eulerian if and only if it has all vertices with even degree.