I just found an old book about number theory and graph theory I used on my first course at university many years ago. Looking inside it, I found a handwritten note pointing to a problem that says:
- Prove that all the $(k-2)$-regular subgraphs of $K_k$ are isomorphic to $K_{k-1}$, where $k$ is an odd integer.
I was thinking about it, but the only thing that came to my mind is the property that says that the sum of the grades of all the vertices equals the double of the number of edges, but I don’t know if this might be useful. I would appreciate any hint to prove that.