# Thread: Find the number of graphs such that...

1. ## Find the number of graphs such that...

Find the number of undirected graphs on $\displaystyle n$ vertices such that, whenever there is a path $\displaystyle a_1 \leftrightarrow a_2 \leftrightarrow ... \leftrightarrow a_k$ joining the vertices $\displaystyle a_1,...,a_k$, then there is a complete graph on $\displaystyle a_1,...,a_k$.

2. Originally Posted by Bruno J.
Find the number of undirected graphs on $\displaystyle n$ vertices such that, whenever there is a path $\displaystyle a_1 \leftrightarrow a_2 \leftrightarrow ... \leftrightarrow a_k$ joining the vertices $\displaystyle a_1,...,a_k$, then there is a complete graph on $\displaystyle a_1,...,a_k$.
I can't think of any hints that wouldn't be completely and utterly cryptic, so I'll just sort of give you the logical leap bit, and is hopefully only a wee bit cryptic...presuming, of course, you didn't quite get the leap bit...

Basically, what you want to do find out how many ways there are of partitioning and n-element set.

I'm sure you can solve the problem from there.