List of ways to tell if degree sequence is impossible for a simple graph I'm trying to make a list of ways to tell if a given degree sequence is impossible. For example $3,1,1$  is not possible because there are only 3 vertices in total so one can't have degree 3.
The list so far


*

*vertices has degree equal to or larger than number of vertices

*sum of degrees is odd

*for n vertices if one has degree n-1 and another has degree 0

*for n vertices the sum of the degrees cannot be greater than $n(n-1)$ because this would be have more edges than a complete graph


The Erdős–Gallai theorem completely characterizes the possible degree sequences for simple graphs.
It is stated by Wikipedia as:

A sequence of non-negative integers $d_1\geq\cdots\geq d_n$ can be represented as the degree sequence of a finite simple graph on $n$ vertices if and only if $d_1+\cdots+d_n$ is even and
  $$ \sum^{k}_{i=1}d_i~\leq~ k(k-1)+ \sum^n_{i=k+1} \min(d_i,k)$$
  holds for $1\leq k\leq n$.

