# Finding the number of unlabeled graphs with $n$ vertices such that each vertex has degree 2

I'm trying to find the number of unlabeled graphs with $$n$$ vertices such that each vertex has degree 2. I know that the edge set for such a graph will have cardinality $$n$$ and that the maximum number of possible edges for any graph with n vertices is $${n}\choose{2}$$ but I'm not exactly sure how to proceed from here.

• Are the vertices labeled or unlabeled? – Carl Schildkraut Sep 21 '18 at 15:32
• @CarlSchildkraut Was literally just about to ask that – Don Thousand Sep 21 '18 at 15:32
• Unlabeled........... – Hai Sep 21 '18 at 15:33

Hint: A graph on $$n$$ vertices with each vertex having degree $$2$$ is simply a graph consisting of disjoint cycles. If the vertices are unlabeled, all you care about are the lengths of each cycle.