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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 $x\inV$ and $\sigma\in\mathcal{S}$. We address the problem of characterising thoseinfinite (simple loopless) digraphs which are generated by finite sets ofderangements. The case of finite digraphs was addressed in earlier work by thesecond and third authors. A criterion is given for derangement generation whichresembles the criterion given by De Bruijn and Erd\H{o}s for vertex colouringsof graphs in that the property for an infinite digraph is determined byproperties of its finite sub-digraphs. The derangement generation property fora digraph is linked with the existence of a finite $1$-factor cover for anassociated bipartite (undirected) graph.

Generating infinite digraphs by derangements