For a complete undirected graph G where the vertices are indexed by [n] = {1,2,3,...,n} where n >= 4
. I am aware that the total number of Hamiltonian Circuits in G is (n-1)! / 2
- If we must traverse the edge
{1,2}
, how many Hamiltonian Circuits are there? - How about if multiple consecutive edges, e.g.
{1,2} {2,3}
must be traversed? - What if multiple non consecutive edges, e.g.
{1,2} {3,4}
must be traversed?
Intuitively, for part 1, the answer seems to be (n-2)! /2
but I am not completely sure. For the other parts, I am completely stumped.
Any help is much appreciated!