Long mutation cycles

A mutation cycle is a cycle in a graph whose vertices are labeled by the quivers in a given mutation class and whose edges correspond to single mutations. For any fixed $n\ge 4$, we describe arbitrarily long mutation cycles involving $n$-vertex quivers. Each of these mutation cycles allows for an arbitrary choice of $\left(\genfrac{}{}{0px}{}{n}{2}\right)$ positive integer parameters. None of the mutation cycles we construct can be paved by short mutation cycles.