I was looking through multiple criteria for Hamiltonian circuits and read several papers such as the following from the university of Manchester. I was particulary intrigued by a theorem Wolframs Math World references as Chvátal's theorem, based on degree sequences, not on closure, but I was unable to find a proof for it.
What I am referring to is the following:
Given a graph $G$'s ordered degree sequence $d_1...d_n$, if for $i<{n\over 2}$ one has that $d_i>i+1$ or $d_{n-i}>n-i$, then $G$ is Hamiltonian.
I have seen it referenced in multiple stack exchange posts, but I have not yet seen a proof for it in the literature. Any reference would be very much appreciated, as would any explanation to the easier Bondy-Chvátal theorem.
Thanks a lot
After a good deal more of research, I indeed found the answer to the question I was looking for. On Science Direct, one can access Chvátals original paper, "On Hamilton's ideals", where a proof of his theorem can be found.
The link to the paper can be found here.