E\in F_k, E^{'}\in F_{k^{'}}, k^{'}<k \ \
\operatorname{and} \ \ E\oplus E^{'}\cong \overline{C_4} \},$$ where $\overline{C_4}$ is a graph with four vertices and four edges, in which two vertices have indegrees two, outdegrees zero, and the other two vertices have indegrees zero, outdegrees two(See Figure 1). The four edges belong to $E$ and $E^{'}$ alternatively. To avoid the confusion with vertices and edges in $G$, we call the $nodes$ and $arcs$ corresponding to $F$ and $K$ in $\Psi$. Observe also that if $(E,E^{'})\in K$ is an arc of $\Psi$, $E^{'}$ can be obtained from $E$ by deleting two edges and adding two others, and that this operation can decrease the number of cycles by one(See Figure 2). Hence every arc $(E, E^{'})$ is directed from a node $E$ in some $F_k$ to a node $E^{'}$ in $F_{k-1}$.

The proof strategy is to define a positive weight function $w$ on the arcs set $K$ such that the total weight of arcs leaving each node $E\in F/F_{1}$ is at least one greater than the total weight of arcs entering $E$. Denote $w^{+}(S)$ and $w^{-}(S)$ the total weight leaving and entering a node set $S$ in $\Psi$ respectively, the strategy ensures $$w^{-}(F_k) + 