Generating digraphs with derangements

Abstract: 

Let S be a collection of derangements (fixed point-free permutations) of a possibly infinite set X. The derangement action digraph DA(X,S) is the digraph on vertex set X that has an arc from x to y if and only if some derangement in S maps x to y. We say that S generates DA(X,S). Derangement action digraphs were introduced by Iradmusa and Praeger in 2019, adapting the definition of a group action digraph due to Annexstein, Baumslag and Rosenberg. 

I will discuss recent work by Iradmusa, Praeger and myself in which we characterise, for each positive integer k, the digraphs that can be generated by at most k derangements. Our result resembles the De Bruijn-Erdős theorem in that it characterises a property of an infinite graph in terms of properties of its finite subgraphs.

This is a seminar of the Combinatorial Mathematics Society of Australasia.

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