Shaking hands problem: 6 people each shaking hands with 3 others

Question

There is a group of 6 people. An arrangement is defined as that each person shakes hands with exactly 3 other people. How many arrangements are there?

Follow up: what if there are $N$ people where $N > 3$?

Answer 1

The first graph is what I would call a "Triangular prism" and the second graph is the complete bipartite graph $K_{3,3}$. I believe these are the only regular graphs on $6$ vertices with valency $3$.