TITLE: A simple requirement for a degree sequence to be graphical
QUESTION [5 upvotes]: The following theorem about the degree sequences of finite simple graphs is quite easy to prove from the Erdos-Gallai theorem.
Let  $0 \lt \alpha \le \beta \lt n$ be integers.   Call $(\alpha,\beta,n)$ paragraphical if every integer sequence  $\alpha \le d_1,\ldots,d_n \le \beta$ with even sum is the degree sequence of a simple graph.
Then $(\alpha,\beta,n)$ is paragraphical iff
$$  n \ge \biggl\lfloor\frac{ (\alpha+\beta+1)^2 }{4 \alpha} \biggr\rfloor.$$
I find it really hard to believe that nobody published this anywhere, but can I find it?  The question is: where is it?
Thanks!

REPLY [5 votes]: Answering my own question:  Almost exactly this result was proved in:
Igor E Zverovich and Vadim E Zverovich. Contributions to the theory of graphic sequences. Discrete Mathematics, 105(1):293–303, 1992.
Having now found this, I recall that someone gave me the tip "Zverovich" at a conference last year, but I lost the tip and also forgot who it was that gave it to me (sorry).
An improvement, allegedly the best possible improvement, appeared in G. Cairns, S. Mendan, Y. Nikolayevsky, A sharp refinement of a result of Zverovich–Zverovich, Discrete Mathematics, Volume 338, Issue 7, 6 July 2015, Pages 1085-1089.
The result of Cairns et al. is similar to what is in my question, but not exactly the same. After looking very closely I can see that my version is wrong. The simplest proof is that it is not closed under complementation. My apologies to anyone who tried to prove it.