Long mutation cycles

Abstract 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 $$n\ge 4$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> </mml:math> , we describe arbitrarily long mutation cycles involving n -vertex quivers. Each of these mutation cycles allows for an arbitrary choice of $$\left( {\begin{array}{c}n\\ 2\end{array}}\right) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mfenced> <mml:mrow> <mml:mtable> <mml:mtr> <mml:mtd> <mml:mi>n</mml:mi> </mml:mtd> </mml:mtr> <mml:mtr> <mml:mtd> <mml:mrow> <mml:mrow/> <mml:mn>2</mml:mn> </mml:mrow> </mml:mtd> </mml:mtr> </mml:mtable> </mml:mrow> </mml:mfenced> </mml:math> positive integer parameters. None of the mutation cycles we construct can be paved by short mutation cycles.