Graphical cyclic permutation groups

Abstract

Abstract. We establish conditions for a permutation group generated by a single permutation of a prime power order to be an automorphism group of a graph or an edge-colored graph. This corrects and generalizes the results of the two papers on cyclic permutation groups published in 1978 and 1981 by

Cyclic Permutation Groups that are Automorphism Groups of Graphs

“…Proof In the case y = z = 1, this is a consequence of Lemma 3.3 in [9]. We have also defined the graph C(x, x) there.…”

“…Also the proof of the main result contained a gap. All this has been corrected in [9]. Our aim is to generalize these results to cyclic permutation groups of arbitrary order.…”

Subgroups of simple primitive permutation groups defined by unordered relations (Automorphism groups of hypergraphs)

“…Based on this, similar results for directed graphs, tournaments, and oriented graphs have been obtained (see e.g., [10]). Partial results have been obtained also for the classes of cyclic permutation groups, abelian permutation groups, and Frobenius groups (see [4,7,13]).…”

Abelian permutation groups with graphical representations

“…The König's problem for this class turned out not so easy as it could seem at the first sight. After some partial results containing errors and wrong proofs ( [22,23] corrected in [10]), the final result has been obtained only recently (announced in [13]). The full description turns out to consist of seven technical conditions concerning possible lengths of the orbits.…”