Graphical representations of cyclic permutation groups

In this paper we prove that a permutation group A generated by a single permutation sigma is an automorphism group of an edge-colored graph if and only if for every orbit O of sigma with cardinality vertical bar O vertical bar >= 3 there exists another orbit O' of sigma such that gcd(vertical bar O vertical bar, vertical bar O'vertical bar) >= 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. (C) 2019 Elsevier B.V. All rights reserved.