the planarity rule, vertex $t+1$ or vertex $t-1$ must also be covered by $\phi(i) \phi(i+1)$. If $\{t+1, t-1 \} = \{ \phi(i), \phi(i+1) \}$, then $n =3$; every spine order of three vertices is a natural ordering, and we obtain a contradiction. Therefore, either $t+1$ or $t-1$ is not in $\{ \phi(i), \phi(i+1) \}$. By continuing this logic for an arbitrary vertex $t$, every vertex except $\phi(i)$ and $\phi(i+1)$ is embedded between $\phi(i)$ and $\phi(i+1)$; otherwise, the planarity rule is violated. By $(2)$, the spine order must be a natural ordering of $V(C)$.

Corollary \[directed cycle\] follows from Theorem \[cyclic\_on\_C\] and provides the location of the arcs in a $1$-page oriented book embedding of a directed cycle.

\[directed cycle\] Let $\overrightarrow{D}_n$ be an $n$-dicycle. Then $\overrightarrow{D}_n$ has a $1$-page oriented book embedding if and only if the spine ordering is a natural ordering of $V(\overrightarrow{C})$ and every arc is tight, except the arc between the top vertex and bottom vertex in the spine.

We now show that every oriented cycle has a $1$-page oriented book embedding.

Let $\overrightarrow{C}$ be an oriented cycle. Then $obt(\overrightarrow{C}) =1$.

Let $C$ be the underlying graph of $\overrightarrow{C}$. Since $bt(C) \geq 1$, we have that 