Derangement action digraphs and graphs

Abstract

A derangement of a set X is a fixed-point-free permutation of X. Derangement action digraphs are closely related to group action digraphs introduced by Annexstein, Baumslag and Rosenberg in 1990. For a non-empty set X and a non-empty subset S of derangements of X, the derangement action digraph DA⃗(X,S) has vertex set X, and an arc from x to y if and only if y is the image of x under the action of some element of S, so by definition it is a simple digraph. In common with Cayley graphs and Cayley digraphs, derangement action digraphs may be useful to model networks since the same routing and communication schemes can be implemented at each vertex. We prove that the family of derangement action digraphs contains all Cayley digraphs, all finite vertex-transitive simple graphs, and all finite regular simple graphs of even valency. We determine necessary and sufficient conditions on S under which DA⃗(X,S) may be viewed as a simple undirected graph of valency |S|. We investigate structural and symmetry properties of these digraphs and graphs, pose several open problems, and give many examples.