Graphical representations of cyclic permutation groups

Abstract

In this paper we prove that a permutation group A generated by a single permutation σ is an automorphism group of an edge-colored graph if and only if for every orbit O of σ with cardinality |O|≥3 there exists another orbit O′ of σ such that gcd(|O|,|O′|)≥3. We prove also that every such permutation group is an automorphism group of a 3-colored graph. This is a step towards a characterization of cyclic permutation groups that are automorphism groups of simple graphs.