If it's worth saying, but not worth its own post (even in Discussion), then it goes here.
Notes for future OT posters:
1. Please add the 'open_thread' tag.
2. Check if there is an active Open Thread before posting a new one. (Immediately before; refresh the list-of-threads page before posting.)
3. Open Threads should be posted in Discussion, and not Main.
4. Open Threads should start on Monday, and end on Sunday.
Your operation for turning G into G' doesn't let you construct all graphs, e.g. K3 (the triangle graph) can't be formed like that. The rest of that paragraph is probably more dense than it needs to be. You're on the right track, but I can't quite tell if you actually rely on that construction.
Thanks for the feedback. I think you can construct all graphs and use it to prove the theorem if you prove that you can add an arbitrary number of additional edges and nodes to an arbitrary graph and keep the sum of the degrees of all nodes even, instead of just one additional node and one additional edge. I also see what you mean about this:
I think the inductive hypothesis in the rest of that paragraph might be enough, and I just wrote down how I intuitively visualized the proof before that ... (read more)