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

TL;DR

This paper proves that the ratios of derangements to permutations in directed graphs can approximate any value in the interval [0, 1/2], showing the set of such ratios is dense within this range.

Contribution

It demonstrates that the set of ratios of derangements to permutations in digraphs is dense in [0, 1/2], answering an open question from prior research.

Findings

Ratios of derangements to permutations are dense in [0, 1/2]

Any value in [0, 1/2] can be approximated by such ratios

The maximum ratio of derangements to permutations in a digraph is 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. 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 $[0, 1/2]$.