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 $f:V \rightarrow V$ 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. Bucic, Devlin, Hendon, Horne and Lund proved that in any digraph, the ratio of derangements to permutations is at most $1/2$. Answering a question posed by Bucic, Devlin, Hendon, Horne and Lund, we show that the set of possible ratios of derangements to permutations in digraphs is dense in the interval $[0, 1/2]$.