Answer
The number of vertices in the complete graph is 6.
Work Step by Step
The number of Hamilton circuits in a complete graph with $n$ vertices is $(n-1)!$.
The complete graph has 120 Hamilton circuits. We can find a number such that the factorial of the number is equal to 120.
5*4*3*2*1 = 120
5! = 120
We can find $n$, the number of vertices in the complete graph.
$(n-1)! = 5!$
$n-1 = 5$
$n = 6$
The number of vertices in the complete graph is 6.
