Abstract
In this paper we establish conditions for a permutation group generated by a single permutation to be an automorphism group of a graph. This solves the so called concrete version of König’s problem for the case of cyclic groups. We establish also similar conditions for the symmetry groups of other related structures: digraphs, supergraphs, and boolean functions.
Highlights
Frucht’s theorem, conjectured by Dénes König states that every abstract finite group is isomorphic to the automorphism group of a graph [30]
The problem asking which permutation groups can be represented as automorphism groups of graphs is known as the concrete version of König’s problem [29]
In [13] we provide a relatively simple characterization of those cyclic permutation groups that are automorphism groups of edge-colored graphs
Summary
Frucht’s theorem, conjectured by Dénes König states that every abstract finite group is isomorphic to the automorphism group of a graph [30]. On the other hand it is known that not every permutation group is an automorphism group of a graph. There is no graph on n vertices whose automorphism group is the cyclic group Cn generated by an n-element cycle. The problem asking which permutation groups can be represented as automorphism groups of graphs is known as the concrete version of König’s problem [29]. This problem turned out much harder and was studied first for regular permutation groups as the problem of Graphical Regular Representation. In [2], Babai uses the result of Godsil to prove a similar characterization in the case of directed graphs. In [20,21], Mohanty et al, consider permutation groups generated by a single permutation (they call them cyclic permutation groups) whose order is a prime or a Supported in part by Polish NCN Grant 2012/07/B/ST1/03318
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