$\phi(i) \rightarrow \phi(i+k)$ $(mod$ $n)$, $i \in [n]$. We call each cyclic permutation of $V(C)$ a [*natural ordering*]{} of $V(C)$.

\[cyclic\_on\_C\] Let $C$ be a cycle having a $k$-page book embedding. Then $k$ is minimal if and only if the spine order is a natural ordering of $V(C)$.

Since $C$ is not a path, by Theorem \[BKthm\], $bt(C) \geq 1$ and since $C$ is outerplanar, $bt(C) \leq 1$. Therefore $k$ is minimal when $k=1$. We need only prove the necessary condition, as the sufficient condition is trivial.

To prove the necessary condition by contradiction, choose a counterexample, that is, suppose that there exists a 1-page book embedding of $C$ such that $(1)$ the spine order is not a natural ordering of $V(C)$; then there exists at least one pair of vertices, $\phi(i)$, $\phi(i+1)$ in $C$ that are not consecutive in the spine, implying that the edge $\phi(i) \phi(i+1)$ is a loose edge. Considering all such pairs, $(2)$ choose $i$ such that no other such pair exists between $\phi(i)$ and $\phi(i+1)$ in the spine. By $(1)$, there is at least one vertex $t$ embedded between $\phi(i)$ and $\phi(i+1)$ in the spine. Since $t$ is covered by the edge $\phi(i) \phi(i+1)$, by 