double of simple paths of lengths $p-1$ and $q-1$, where $p,q \geq
2$. For the case $p\neq q$ there are at most $n_i$ ways such that $\gamma_{i}\oplus\gamma$ is a pair of cycles with length $p$ and $q$, and $n_i/2$ ways such that $\gamma_{i}\oplus\gamma$ is a pair of cycles with length $p$ and $q$ for the case $p=q$. Noting both cases happen when $\gamma_i$ is contained in a complete sub digraph of $G$ or $G$ is a complete digraph(Complete digraph is defined as such a digraph that any two distinct vertices have edges pointing to each other). Hence\
$$\begin{split}
w^{-}(E)&=\sum\limits_{E^{'}:(E^{'},E)\in K}w(E^{'},E)\\
&\leq \sum\limits_{i=1}^k n_i f(k)\sum\limits_{\substack{p>q\geq
2\\p+q=n_i}}(\frac{1}{p}+\frac{1}{q})+\frac{1}{2}\sum\limits_{i=1}^k
n_i f(k)\sum\limits_{\substack{p=q\geq
2\\p+q=n_i}}(\frac{1}{p}+\frac{1}{q})\\
&\leq \frac{1}{2}\sum\limits_{i=1}^k n_i
f(k)\sum\limits_{\substack{p,q\geq
2\\p+q=n_i}}(\frac{1}{p}+\frac{1}{q})\\
&=\frac{1}{2}\sum\limits_{i=1}^k n_i f(k)\sum\limits_{p=2
}^{n_i-2}(\frac{1}{p}+\frac{1}{n_i-p})\\
&=\sum\limits_{i=1}^k n_i f(k)\sum\limits_{p=2
}^{n_i-2}(\frac{1}{p})\\
&\leq\sum\limits_{i=1}^k n_i f(k)\log n_i\\
&\leq n \log n f(k). \ \ \ \Box
\end{split}$$\
\
\
*4.2. Hardness of Counting Hamiltonian cycles in dense digraphs*\
We first declare the notation related to undirected graphs only appears in this subsection and the notation related to digraph is the same as that in the previous sections. Our reduction comes from the undirected graph, hence notation for undirected graphs is needed. Let $G$ be a simple $undirected$ graph with vertices $\{1,2,\cdots,n\}$, where 