  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.
