A sequence of non-negative integers is called graphic if there exist a graph whose degree sequence is precisely that sequence.
For example, $(1, 1, 1, 1, 1, 1)$ is a graphic since it's the degree sequence of graph
and $(3, 1, 0, 0, 0)$ is not a graphic since there is no graph with one vertex of degree 3, one vertex of degree 1, and three vertices of degree 0.
Now, the question is prove that
$$(n, n, n-1, n-1, \ldots, 4, 4, 3, 3, 2, 2, 1, 1)$$
is always a graphic.
My attempt is by construct the simple graph and find some pattern for building a larger one. Starting with $n=1$, then I try to construct $n=2$. Now, based on the result $n=2$, then again I construct for $n=3$, and so on.
But, after doing that process, I don't get at all the pattern for building such larger graph from previous graph. Everything seems randomly yet beautifully constructed. Here is my drawing process:
Please help me to prove this. Thanks before.