Suppose you have a given degree sequence $(d_1,d_2,\dots,d_n)$, where $d_i$ is even for every $i$. Does there exist a general graph with this degree sequence?
I say yes, the easiest way is to take any isolated graph of $n$ vertices, $\{v_1,\dots,v_n\}$, and then at each $v_i$, put in $d_i/2$ loops, so $\deg(v_i)=d_i$ for all $i$. Is this cheating? It seems very easy, and it all hinges on the fact that graph is allowed to have multiple edges between vertices.
As a follow up question, is there some way, sufficient and/or necessary conditions to tell if a graph with a given degree sequence exists, given that the sum of the degree sequence is even? (It may not necessarily be the case that every degree may be even itself.)