On the Number of Hamiltonian Cycles in Hamiltonian Dense Graphs

Let $G$ be a Hamiltonian graph with $n$ vertices and $Cn(n-1)/2$ edges, where $3/4<C\le 1$. We show that $G$ contains at least $(C_1n)^{C_2n}$ Hamiltonian cycles, where $C_1$ and $C_2$ are some constants depending on $C$, and prove an analog of Dirac's theorem for graphs with prescribed edges.