Rotational circulant graphs

TL;DR

This paper classifies first-kind Frobenius circulant graphs that admit complete rotations, explores their construction, and provides counterexamples to a conjecture about fixed-point sets in Cayley graphs.

Contribution

It provides a complete classification of Frobenius circulant graphs with complete rotations and introduces new constructions and counterexamples related to Cayley graph automorphisms.

Findings

Classified all first-kind Frobenius circulant graphs admitting complete rotations.

Established a necessary and sufficient condition for such graphs to be 2-cell embeddable as balanced Cayley maps.

Constructed non-Frobenius circulants with fixed-point sets that are independent and not vertex-cuts.

Abstract

A Frobenius group is a transitive permutation group which is not regular but only the identity element can fix two points. Such a group can be expressed as the semi-direct product $G = K \rtimes H$ of a nilpotent normal subgroup $K$ and another group $H$ fixing a point. A first-kind $G$-Frobenius graph is a connected Cayley graph on $K$ with connection set an $H$-orbit $a^H$ on $K$ that generates $K$, where $H$ has an even order or $a$ is an involution. It is known that the first-kind Frobenius graphs admit attractive routing and gossiping algorithms. A complete rotation in a Cayley graph on a group $G$ with connection set $S$ is an automorphism of $G$ fixing $S$ setwise and permuting the elements of $S$ cyclically. It is known that if the fixed-point set of such a complete rotation is an independent set and not a vertex-cut, then the gossiping time of the Cayley graph (under a certain model) attains the smallest possible value. In this paper we classify all first-kind Frobenius circulant graphs that admit complete rotations, and describe a means to construct them. This result can be stated as a necessary and sufficient condition for a first-kind Frobenius circulant to be 2-cell embeddable on a closed orientable surface as a balanced regular Cayley map. We construct a family of non-Frobenius circulants admitting complete rotations such that the corresponding fixed-point sets are independent and not vertex-cuts. We also give an infinite family of counterexamples to the conjecture that the fixed-point set of every complete rotation of a Cayley graph is not a vertex-cut.