Generating infinite digraphs by derangements

TL;DR

This paper characterizes infinite digraphs generated by finite sets of derangements, linking their structure to properties of finite sub-digraphs and bipartite graph covers, extending previous finite cases.

Contribution

It provides a new criterion for derangement generation of infinite digraphs, connecting it with finite sub-digraph properties and bipartite graph 1-factor covers.

Findings

Criterion for derangement generation in infinite digraphs.

Connection between derangement generation and bipartite graph 1-factor covers.

Extension of finite digraph results to infinite cases.

Abstract

A set $\mathcal{S}$ of derangements (fixed-point-free permutations) of a set $V$ generates a digraph with vertex set $V$ and arcs $(x,x^\sigma)$ for $x\in V$ and $\sigma\in\mathcal{S}$. We address the problem of characterising those infinite (simple loopless) digraphs which are generated by finite sets of derangements. The case of finite digraphs was addressed in earlier work by the second and third authors. A criterion is given for derangement generation which resembles the criterion given by De Bruijn and Erd\H{o}s for vertex colourings of graphs in that the property for an infinite digraph is determined by properties of its finite sub-digraphs. The derangement generation property for a digraph is linked with the existence of a finite $1$-factor cover for an associated bipartite (undirected) graph.