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

Abstract

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 a permutation that does not fix any vertex. In [1] it is proved that in any digraph, the ratio of derangements to permutations is at most $1/2$. Answering a question posed in [1], we show that the set of possible ratios of derangements to permutations in digraphs is dense in the interval $[0,1/2]$.