TITLE: handshaking lemma and Erdos-Gallai theorem
QUESTION [7 upvotes]: The conditions for a sequence to be the degree sequence of a simple graph are given by the Erdos-Gallai theorem in addition to the handshaking lemma. Is there an example of a degree sequence where the handshaking lemma is satisfied, but the Erdos-Gallai theorem is not satisfied and thus, the sequence is not graphic (or vice versa).
Or, does satisfying one of these conditions ensure the other is always satisfied?

REPLY [2 votes]: The simplest example would be
$$
(2,0,0)
$$
The handshaking lemma is satisfied, but the Erdős–Gallai inequalities are not, as $2\not\le 1\cdot (1-1)+0+0$.