alt Hacker Newsllms are good at greedy depth first search[1]. so prompt / human needs to tell the model the breadth parts to take.
and human writing the prompt here did a lot more than that. asking to include parallel-edge 2-cycles, admitting disconnected graphs, specifying the emptyset cover, multiset counting, it all likely came from previous fake proofs.
[1]: see: ankitmaloo.com/fable - its the way most llms are trained, and is also natural owing to autoregressive nature.
I guess "depth first" is just an expression of so called chain of thought, which is just a linear sequence. I don't know if GPT has any search-like (tree search) algorithms in their reasoning, it would be quite interesting if they did (they probably have researched this area, at least).