Question

How many Hamiltonian circuits are there in a complete graph with 8 vertices?
?Notes: (The answer is not 7 and the answer is not 5040)

Answer 1

I mean for n vertices, I choose any 2 vertices (that's an edge) and for each other vertex by connecting from each vertex from my edge by new edges, I can create a triangle, which is a Hamiltonian circle of size 3 and so on. But there are a lot of repeats and that's a mess.

we know that circular permutations there fore we have

(n?1)! is the possibilities to order n different elements in a circle.

also each vertex degree is 2

hence (n-1)!/2

therefore (8-1)!/2 = 2520