The set of ratios of derangements to permutations in digraphs is dense in $[0,1/2]$

Bethany Austhof, Patrick Bennett, Nick Christo

A permutation in a digraph $G=(V,E)$ is a bijection $f:V\to V$ such that for all $v\in V$ we either have that $f$ fixes $v$ or $(v,f(v))\in E$. A derangement in $G$ is…