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arXiv · 2101.02995

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

Bethany Austhof·Patrick Bennett·Nick Christo
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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 $[0, 1/2]$.

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Bethany Austhof, Patrick Bennett, Nick Christo. 2021-01-08. The set of ratios of derangements to permutations in digraphs is dense in $[0, 1/2]$. https://arxiv.org/abs/2101.02995

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