# A simple requirement for a degree sequence to be graphical

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!

• I take it paragraphical is your term and not standard. Any idea what term might be used, or who might have worked on ranges of degree sequences? Gerhard "Not On My Tongue Tip" Paseman, 2017.01.29. Jan 30 '17 at 6:13
• @GerhardPaseman: I just made it up for the question. I doubt if there is a standard term. Lots of people worked on degree sequences. Jan 30 '17 at 6:47