Does there exist a $10$-vertex graph with degrees $1, 1, 3, 3, 6, 6, 6, 7, 8, 9$? How to tell if there exists a graph given degrees of vertices? Not sure how to go about this one
Is there a graph on $10$ vertices with degrees $1, 1, 3, 3, 6, 6, 6, 7, 8, 9$?

You can solve such problems with the Havel-Hakimi Theorem. The theorem states that either all the following degree sequences can be realized as a simple graph or none of them can. To get the next sequence in the list, remove the first element (call it $k$) and reduce the next $k$ elements by one (assuming always that the sequence is written in non-increasing order).


*

*9, 8, 7, 6, 6, 6, 3, 3, 1, 1

*7, 6, 5, 5, 5, 2, 2, 0, 0

*5, 4, 4, 4, 1, 1, -1, 0


We can go ahead and stop because no vertex can have degree of $-1$. By the theorem, your original sequence cannot be realized either.
