For a finite group G and a subset S of G, the Cayley digraph 
is the directed graph with vertex set G and arcs from 
to 
for each 
and
. The set S is often called the connection set of the digraph
, and this digraph is connected if and only if 
is a generating set for G. The connection set S is said to be minimal if it is a minimal generating set of G, and it is said to be minimum if it is a minimal connection set of minimum cardinality. A Hamilton cycle (path) in a digraph of with n vertices is a directed cycle (path) with n vertices. A digraph is said to be Hamiltonian if it has a Hamilton cycle.
Each arc in 
of the form 
is labelled
, and called an s-arc. A Hamilton cycle in 
can be specified by the sequence of vertices encountered or by the sequence of arcs traversed. In the latter case, it is often more convenient to list the labels of the arcs, rather than the arcs themselves, since for each vertex there is exactly one out-arc with label 
for each
. An ordered sequence 
of the arc labels encountered in a Hamilton cycle is called a Hamiltonian arc sequence. Since Cayley digraphs are vertextransitive, any cyclic shift of a Hamiltonian arc sequence of a Cayley digraph is also a Hamiltonian arc sequence of the digraph, and traversing a Hamiltonian arc sequence of a Cayley digraph starting from any vertex will yield a Hamilton cycle of the digraph. For convenience and brevity, we sometimes omit the commas and brackets from an arc sequence. For an arc sequence x, the symbol 
denotes the concatenation of 
copies of
. If 
for some 
and
, we sometimes write
to denote the fact that there is an s-arc from 
to 
in
.
For an integer
, the symbol 
denotes the dihedral group of order
. For 
this is the group of symmetries of the regular 
-gon under the operation of function composition, and it has the presentation
, where R is the counterclockwise rotation of 
and 
is a reflection across any axis of symmetry. For n = 2 the same presentation can be used to define D4. Note that
.
One fundamental problem is that of determining which Cayley digraphs are Hamiltonian. This is a longstanding problem which can be traced back to bell ringing, or campanology, since the orders in which a set of church bells may be rung form a group, and a Hamilton cycle in a Cayley digraph of this group gives a sequence of these bell ringing orders which is pleasing to the ear. The problem is longstanding mainly due to its difficulty. There are several good surveys on the problem, including [1-3], which discusses recent progress and current directions in the more general related problem of finding Hamilton cycles and paths in vertex-transitive graphs.
One of the first elegant results on the problem of the Hamiltonicity of Cayley digraphs is due to Rankin [
with connection set 
are Hamiltonian, in the case where 
is a normal subgroup of
.This solves the problem for Cayley digraphs with two generators for a class of groups which includes the Abelian groups, and some Cayley digraphs on solvable groups with two generators. In Section 2 we prove the following theorem.
Theorem 1.1. A Cayley digraph on 
with outdegree two is Hamiltonian if and only if it is connected.
This is a new result since such digraphs do not satisfy the hypothesis of Rankin’s result. We first prove that if 
is generated by two elements then both 
and m are odd, and the proof makes use of the following result due to Gaschütz in 1955 [
Proposition 1.2. (Gaschütz [
and 
be groups. If 
is finite, then 
is generated by two elements if and only if each of the groups 
is generated by two elements, where 
is the intersection of the maximal proper normal subgroups of 
for
.
is generated by two elements, then both 
and 
are odd.
is generated by two elements if and only if 
is generated by two elements, where 
and 
denote the intersections of the maximal normal subgroups of 
and
, respectively. If 
is even and
, then the normal subgroups of
is
. On the other hand, if 
is odd, then the normal subgroups of 
are all of the form 
for
, and so in this case there is only one maximal normal subgroup, namely
. Note that 
and 
. Thus Proposition 1.2 implies that if 
and 
are both even then 
is generated by two elements if and only if 
is generated by two elements, and if exactly one of n or m is even, say n is even, then 
is generated by two elements if and only if 
is generated by two elements. Using induction and the fact that
, we conclude that if 
or 
is even and 
is generated by two elements, then either 
or 
is generated by two elements, a contradiction. Hence both 
and 
must be odd.
of outdegree two is connected, then 
is generated by two elements. Let 
be a generating set for
. By Lemma 2.1, both 
and 
must be odd. If 
and 
are both rotations in 
for some
, then S does not generate
, a contradiction. Hence at least one of 
and 
is a reflection for
. If 
are all reflections, then every element in 
is an ordered pair of reflections or an ordered pair of rotations, a contradiction. If a rotation in 
does not generate the cyclic subgroup of rotations of
, or if a rotation in 
does not generate the cyclic subgroup of rotations of
, then S does not generate all of
, a contradiction. Thus any 2-element generating set 
must have one of the following two forms:
for reflections 
and
, and rotations 
and 
or orders 
and
, respectively.
. Each element of 
may be written uniquely in the form 
where
, 
, and
. For convenience, we will represent the element 
by the ordered string
. Following the arc labelled 
from a given vertex 
of the digraph increases the value of 
by 1 modulo 
if
, decreases the value of 
by 1 modulo 
if
, fixes the value of
, increases the value of 
by 1 modulo 2, and fixes the value of
. Similarly, following the arc labelled 
from a given vertex 
of the digraph increases the value of 
by 1 modulo 
if
, decreases the value of 
by 1 modulo 
if
, fixes the value of
, increases the value of 
by 1 modulo 2, and fixes the value of
.
, we form a path which visits each vertex of the form
, for which 
and 
have the same parity, exactly once. Now following 
we extend this path to visit each vertex of the form 
where
. Again following
, we visit each vertex of the form 
where
. Continuing in this way, starting from 0000 and following the arc sequence
, we form a path which visits each vertex of the form 
where both 
and
. Starting from the last vertex on this path and following the arc sequence
, we visit each vertex of the form 
where 
and
. In total, starting from the identity vertex 0000 and following arc sequence
, we form a path which visits each vertex of the form 
exactly once, where
. Now following arc 
from the last vertex on this path, we land on the first vertex 
of our path of the form 
with
. Now repeating the arc sequence
, we visit each vertex of the form 
with 
exactly once, and finish on the vertex
. Thus we have formed a Hamilton path. Finally, following arc
, we land back on the identity vertex 0000. Hence the complete arc sequence
is a Hamiltonian arc sequence. This arc sequence is shown in 
and
. 
for reflections 
and
, and rotations 
and 
or orders 
and
, respectively.
. First we will prove that this arc sequence traces out a walk in 
which visits all vertices, then we will show that this walk is closed. Note that each vertex of this graph can be written uniquely in the form 
where
, 
and
.
visits all vertices of the digraph, notice that starting at any vertex 
and following arc sequence
, we visit all vertices in the coset of 
which contains
. Hence, starting from
, if the vertex 
reached by arc sequence 
and the vertex 
reached by arc sequence 
lie in different cosets of 
whenever
, we can conclude that the arc sequence 
traces out a walk which visits all vertices of the digraph. Suppose, for the sake of contradiction, that 
and 
lie in the same coset of
. We have
-arcs doesn’t change the exponent of
. Also, each time a vertex travels by an a-arc, its first coordinate alternates between 
and
, and each time a vertex travels by the arc sequence
, its second coordinate alternates between 
and
. Hence
.
and 
are in the same coset of
, 
can be reached from 
through a sequence of 
-arcs, which does not change the exponent of 
in the second coordinate. Thus
in 
and 
are equal, 
can be reached from 
by a sequence of aarcs of even length. This implies that
, this implies
, so
is odd we have
and 
lie in different cosets of
, and so the walk traced by the arc sequence 
visits every vertex of the digraph.
and observe that
traces out a walk of length 
which is closed and visits every vertex of the digraph, we conclude that it is a Hamiltonian arc sequence. This arc sequence is shown in 
and
.