Hamiltonian Cayley Digraphs on Direct Products of Dihedral Groups *

We prove that a Cayley digraph on the direct product of dihedral groups D2n × D2m with outdegree two is Hamiltonian if and only if it is connected.


Introduction 1.Definitions
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 v to vs for each v and ( ; ) Cay G S G  s S  .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.

Cay G S ; ) S
Each arc in of the form is labelled ( ; ) Cay G S ( , ) v vs s , 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 ( ) Cay G;S s s S  .An ordered sequence 1 2  , , ,  n s s  s 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 t x denotes the concatenation of copies of t x .If vs w  for some and , v w G  s S  , we sometimes write to denote the fact that there is an s-arc from to w in .
For an integer , the symbol 2n 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 360 n  and F is a reflection across any axis of symmetry.For n = 2 the same presentation can be used to define D 4 .Note that .

History and Layout of the Paper
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 [4] 2 n m 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 [5].

G G is finite, then G 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 .

Direct Products of Dihedral Groups
In this section we prove Theorem 1.1.We make use of the following lemma.
Lemma 2.1.If 2 is generated by two elements, then both and are odd.
Since any dihedral group is generated by two elements, Proposition 1.2 implies that is generated by two elements if and only if is generated by two elements, where and 2m denote the intersections of the maximal normal subgroups of 2n and 2m , respectively.If n is even and , then the normal subgroups of Thus the intersection of the maximal normal sub- . On the other hand, if is odd, then the normal subgroups of 2n are all of the form for 1 , and so in this case there is only one maximal normal subgroup, namely , and rotations R  and 0 , < 2 j   . For convenience, we will represent the Starting for the identity vertex 0000 and following the sequence  , 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 , and rotations It is easy to see that traveling by a sequence of -arcs doesn't change the exponen F .Also, eac e a vertex travels by an a-arc, its first coordinate alternates be h tim tween ( 1) mod 2 ( ) mod , odd , , even , j vert then we will show that this walk is closed.Note that each vertex o tten uniquely in the form vertices in the coset of a which contains from v , if the vertex v .x and y lie in the same coset of a .We have

Figure 1 .Figure 2 .Finally
Figure 2. Hamilton cycle in the Cayley digraph on D 10 × D 6 with connection set .The label ijkl denotes the vertex   , i j k l F R F R 5 5 3 3 .      2 2 1