(In this representation of $\phi$, the first row specifies the edges and the second row specifies the two vertices of that edge)
I tried and actually draw some example graphs for $\phi$ and I can conclude that the graph is NOT complete. If it was I could use the theorem that says that a complete graph has $\frac{(n-1)!}{2}$ Hamiltonian cycles.
What can I do now since it's not a complete graph?