Is a graph simple, given the number of vertices and the degree sequence? Does there exist a simple graph with five vertices of the following degrees?
(a) 3,3,3,3,2
I know that the answer is no, however I do not know how to explain this.
(b) 1,2,3,4,3
No, as the sum of the degrees of an undirected graph is even.
(c) 1,2,3,4,4
Again, I believe the answer is no however I don't know the rule to explain why.
(d) 2,2,2,1,1
Same as above.
What method should I use to work out whether a graph is simple, given the number of vertices and the degree sequence? 

The answer to both a, and d, is that in fact such graphs exit. It is not hard to find them.
The answer for c is that there cannot be such a graph - since there are 2 vertices with degree 4, they must be connected to all other vertices. Therefore, the vertex with degree one, is an impossibility.
