methodology is analogous to the approach for undirected graphs given by Dyer et.al.[@DFJ94].\
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**Lemma 8.** ([@DFJ94]) *Let $n$ be a natural number and $\beta$ a positive number. Let $k_0 = \max(\lfloor\beta\log
n\rfloor,1)$ and $g(k)=n^{\beta}k!(\beta\log n)^{-k}$, define\
$$f(k)=\left\{ \begin{aligned}
 g(k),\ \ \ \ k\leq k_0 \\
 g(k_0),\ \ k>k_0.
\end{aligned} \right.$$ Then $f(k-1)\geq (\beta\log n)k^{-1}f(k)$; and $f(k)\geq 1$ for any $k$.*\
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**Proof.** If $k\leq k_0$, $f(k-1)=g(k-1)=(\beta\log
n)k^{-1}g(k)=(\beta\log n)k^{-1}f(k)$;

If $k> k_0$, then $\beta\log n/k\leq 1$. Hence $$f(k-1)=g(k_0)\geq(\beta\log n)k^{-1}g(k_0)=(\beta\log
n)k^{-1}f(k).$$ Thus $f(k)\geq f(k_0)$, we have\
$$\frac{1}{f(k)}\leq \frac{1}{f(k_0)}\leq\frac{(\beta \log
n)^{k_0}}{n^{\beta}(k_0)!}\leq
n^{-\beta}\sum\limits_{k=0}^{\infty}\frac{(\beta \log
n)^{k}}{(k)!}\leq n^{-\beta}e^{\beta \log n}=1.\ \ \ \Box$$\
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**Theorem 9.** *Suppose $\alpha\in(.75,1]$. Let $G$ be an $\alpha n$ dense digraph and $F_k$ the set of 1-factors in $G$ containing exactly $k$ cycles, $1\leq k\leq \lfloor n/2\rfloor$. Note that $F_1$ is the set of Hamiltonian cycles in $G$. Let $F=\bigcup_kF_k$. Then $$\frac{|F|}{|F_1|}=O(n^{1+1/(2\alpha-1.5)}).$$*

With this theorem, we prove the main result of this paper Theorem M.\
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**Proof of Theorem M.** By theorem 9, since $|F|/|F_1|=\operatorname{per}(A_G)/\operatorname{ham}(A_G)$, and noting (1), therefore Theorem M follows immediately.\

![](newfigure1.eps)

![](newfigure2.eps)

Now we proceed to prove Theorem 9.\
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**Proof of Theorem 9.** We construct a new weighted digraph $\Psi=(F,K)$. $K$ is defined as follows. $$K=\{(E,E^{'}): 