General Cyclic Orthogonal Double Covers of Finite Regular Circulant Graphs ()
1. Introduction
All graphs we deal with are undirected, finite and simple. Let 
be any regular graph, and let 
be a collection of 
subgraphs (pages) of
. The collection 
is an orthogonal double cover (ODC) of 
if it has the following properties:
1) Double cover property:
Every edge of 
is contained in exactly two of the pages in
.
2) Orthogonality property:
For any two distinct pages 
and
, 
if and only if i and j are adjacent in
.
If all pages 
for all 
then 
is an ODC of 
by
. An automorphism of an ODC 
of 
is a permutation 
such that 
where for 
is a subgraph of 
with 
and 
According to the obvious properties of ODCs by a graph
, the underlying graph H has to be 
-regular. This concept is a generalization of the definitions of an ODC of complete graphs and complete bipartite graphs, which has been studied extensively [1] -[2] . El-Shanawny et al. studied extensively the ODC of complete bipartite graphs; see [3] -[6] . An effective method to construct ODCs in the above cases was based on the idea of translate a given subgraph 
by a group acting on 
If the cyclic group of order 
is a subgroup of the automorphism group of 
(the set of all automorphisms of
), then an ODC 
of 
is cyclic (CODC). Therefore, the circulant graph is of special interest. In [7] , Scapellato et al. offers some insights on the case on ODC of Cayley graphs on cyclic groups. In [8] , Hartmann and Schumacher proved the following: 1) Let 
be a 2-regular graph. There exists an ODC of 
by 
with three exceptions for
: 
and
, 2) Let 
be a 3-regular graph containing a 1-factor and without a component isomorphic to
. There exists an ODC of 
by
, 3) Let 
be a 3-regular graph containing a 1-factor and
. There exists an ODC of 
by
. In [9] , Sampathkumar et al. introduced a special kind of orthogonal labelling called orthogonal σ-labelling, and they found it for some caterpillars of diameters 4. In [7] , Scapellato et al. studied the ODC of Cayley graphs and proved the following: 1) All 3-regular Cayley graphs, except
, have ODCs by
, 2) All 3-regular Cayley graphs on Abelian groups, except
, have ODCs by
3) All 3-regular Cayley graphs on Abelian groups, except 
and the 
- prism (Cartesian product of 
and
), have ODCs by 
In [10] , Sampathkumar et al. completely settled the existence problem of CODCs of 4-regular circulant graphs.
The above results on ODCs of graphs with lower degrees motivate us to consider CODCs of graphs with higher degrees. In [11] , El-Shanawny et al. deal with cayley graphs on abelian groups and proved the existence of ODCs of cayley graphs by several classes of graphs. Here we are concerned with CODCs of circulant graphs of finite degrees higher than
. The paper is organized as follows, Section 1.1 describes the method that can be used throughout. Section-2 constructs CODCs of circulant graphs of finite degrees higher than 
by certain graph classes. Section 3 offers the general CODCs of circulant graphs.
Definition 1. For a sequence 
of positive integers with
, the circulant graph
, has vertex set
; two vertices 
and 
are adjacent, if and only if 
for some
, 
For an edge 
in
, the length of 
is 
Given two edges 
and 
of the same length 
in
, the rotation distance 
between 
and 
is 
where addition and difference are calculated inside 
Note that if 
then the edges 
and 
are adjacent; if 
then the edges 
and 
are non adjacent.
Throughout the paper we make use of the usual notation: 
for the complete graph on 
vertices, 
for the complete bipartite graph with independent sets of sizes 
and
, 
for the path on 
vertices, 
for the cycle on 
vertices, 
for the disjoint union 
of 
and
, 
for 
disjoint copies of 
and 
for the union of 
and 
with a common vertex 
belongs to 
and 
Let 
be positive integers, 
and 
for 
the caterpillar
is the tree obtained from the path 
by joining vertex 
to 
new vertices, 
Other terminology not defined here can be found in [12] .
CODC of Circulant Graphs
Consider the complete graph 
The authors of [13] introduced the notion of an orthogonal labelling. Given a graph 
with 
edges, a 
mapping 
is an orthogonal labelling of 
if the following conditions are satisfied:
1) For every
, 
contains exactly two edges of length
, and exactly one edge of length 
if 
is even, and 2) For every 
The following theorem of Gronau et al. [13] relates CODCs of 
and orthogonal labellings.
Theorem 2. ([13] ) A CODC of 
by a graph 
exists if and only if there exists an orthogonal labelling of
.
Sampathkumar and Srinivasan [10] , called an orthogonal 
-labelling and generalized it to an orthogonal 
-labelling, where 
is a sequence of positive integers with 
.
1) Either n is odd or even and 
Given a subgraph 
of 
with 
edges, a labelling of 
in
, is an orthogonal 
-labelling of 
if:
a) For every
, 
contains exactly two edges of length
, and b) 
2) 
is even and 
Given a subgraph 
of 
with 
edges, a labelling of 
in
, is an orthogonal 
-labelling of 
if:
a) For every
, 
contains exactly two edges of length
, and 
contains exactly one edges of length
, and b)
, The following theorem of Sampathkumar and Simaringa [10] , is a generalization of Theorem 2.
Theorem 3 ([10] ). A CODC of 
by a graph 
exists, if and only if there exists an orthogonal 
-labelling of 
2. CODCs of Circulant Graphs by Certain Graph Classes
This section is devoted to constructing the cyclic orthogonal double covers (CODCs) of circulant graphs by different classes of graphs, complete bipartite graph as in Section 2.1, the union of the co-cycles graph with a star, the center vertex of which, belongs to the co-cycles graph as in Section 2.2 and graphs that are connected by a one vertex as in Section 2.3.
2.1. CODCs by a Complete Bipartite Graph
Theorem 4. For any positive integers 
such that
, there exists a CODC of 
-regular 
by
.
Proof Let us define 
by 
for 
and 
for
Then the edges of length 
where 
where 
is defined for
. For every 
contains exactly two edges of length
, and 
, and hence 
has an orthogonal labelling. By Theorem 3, there exists a CODC Of 
-regular 
by
.
Let us define 
to be the co-cycles graph (the union of 
cycles of length 
with a one vertex 
in common). In the following section we construct a CODCs of finite regular circulant graphs by 
(the union of co-cycles graph with a star whose center vertex is the vertex
).
2.2. CODCs of Circulant Graph by 
Theorem 5 For any positive integer 
there exists a CODC of 
-regular 
by 
Proof Let us define 
by 
for 
where, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
and
. Then the edges of length 
are 
and
; those of length 
are 
and 
those of length 
are 
and 
those of length 
are 
and
; those of length 
are 
and 
those of length 
are 
and 
those of length 
are 
and 
those of length 8 are 
and 
the edges of length l where 
are 
and 
For every 
contains exactly two edges of length 
and 
and then 
and hence 
has an orthogonal labelling. By Theorem 3, there exists a CODC of 
-regular 
by 
for 
Theorem 6 For any positive integer
, there exists a CODC of 
-regular 
by
.
Proof Let us define 
by 
for 
where
, 
, 
and
. Then the edges of length 
are 
and 
those of length 
are 
and 
; those of length 
are 
and 
the edges of length 
where
are 
and
. For every
, 
contains exactly two edges of length
, and since every two edges of the same length are adjacent then 
and hence 
has an orthogonal labelling. By Theorem 3, there exists a CODC of 
-regular 
by 
for
.
Theorem 7 For any positive integer
, there exists a CODC of 
-regular 
by
.
Proof Let us define 
by 
for
, 
, 
, 
, 
Then the edges of length
are 
and 
those of length 
are 
and 
those of length 3 are 
and 
the edges of length 
where 
are 
and 
For every 
contains exactly two edges of length
, and since every two edges of the same length are adjacent then 
and hence 
has an orthogonal labelling. By Theorem 3, there exists a CODC of 
-regular 
by 
for
.
Theorem 8 For any positive integer 
there exists a CODC of 
-regular 
by
.
Proof Let us define 
by 
for
,
, 
, 
and 
Then the edges of length 
are 
and 
those of length 
are 
and 
those of length 
are 
and 
those of length 
are 
and 
the edges of length 
where 
are 
and
. For every 
contains exactly two edges of length
, and since every two edges of the same length are adjacent then 
and hence 
has an orthogonal labelling. By Theorem 3, there exists a CODC of 
-regular 
by 
for
.
Theorem 9 For any positive integer 
there exists a CODC of 
-regular 
by
.
Proof Let us define 
by 
for
, 
, and
. Then the edges of length 
are 
and 
those of length 
are 
and 
those of length 
are 
and 
those of length 
are 
and 
those of length 
are 
and 
the edges of length 
where 
are 
and 
For every 
, 
contains exactly two edges of length
, and since every two edges of the same length are adjacent then 
and hence 
has an orthogonal labelling. By theorem 3, there exists a CODC of 
-regular 
by 
for
.
According to these results, we can pose the following conjecture:
Conjecture 1. For any positive integers 
such that 
and
, there exists a CODC of 
-regular 
by
.
2.3. CODCs by 
Graphs that Are Connected by a One Vertex a
Theorem 10 For any positive integer 
there exists a CODC of 
-regular 
by
.
Proof Let us define 
by 
for
, 
, 
, 
and
. Then the edges of length 
are 
and
; those of length 
are 
and
; those of length 
are 
and 
those of length 
are 
and 
the edges of length 
where 
are 
and 
For every 
contains exactly two edges of length
, and since every two edges of the same length are adjacent then 
and hence 
has an orthogonal labelling. By Theorem 3, there exists a CODC of 
-regular 
by 
for
.
Theorem 11 For any prime number 
there exists a CODC of 
-regular 
by
.
Proof Let us define 
by 
for
, 
, 
, 
and
. Then the edges of length 
are 
and 
those of length 
are 
and
; those of length 
are 
and 
the edges of length 
where 
are 
and
. For every 
contains exactly two edges of length
, and since every two edges of the same length are adjacent then 
and hence 
has an orthogonal labelling. By Theorem 3, there exists a CODC of 
-regular 
by 
for
.
Theorem 12 For any positive integer 
there exists a CODC of 
-regular
by
.
Proof Let us define 
by 
for
, 
, 
, 
, 
and 
Then the edges of length 
are 
and 
those of length 
are 
and 
those of length 
are 
and 
the edges of length 
where 
are 
and 
For every 
contains exactly two edges of length
, and since every two edges of the same length are adjacent then 
and hence 
has an orthogonal labelling. By Theorem 3, there exists a CODC of 
-regular 
by 
for
.
Conjecture 2. For any positive integers 
so that
, there exists a CODC of 
-regular 
by
.
3. General CODCs of Circulant Graph
In constructing CODCs a natural approach is to try to use given CODCs to obtain CODCs of a larger Circulant Graph. That is we will do in the following theorem.
Theorem 13 For any positive integers 
if there exists a CODC of 
by G with respect to 
Then there exists a CODC of 
where 
by 
with respect to
.
Proof Let the 
has a CODC by 
with respect to
. Then the graph 
has an orthogonal 
-labelling with respect to
. And hence, for every
, 
contain exactly two edges of the length 
as 
and 
where 
and
. And 
to construct a CODC of 
for 
by 
with respect to
, the graph 
must have an orthogonal 
-labelling with respect to
. From the orthogonal labelling of 
we can obtain an orthogonal labelling of 
as follows Case 1: Either 
is odd or even and 
where 
For every 
contain exactly two edges of the length 
as 
and 
where
and
By the definition of
,
and
, we have 
Then the graph 
has an orthogonal 
-labelling with respect to
.
Case 2: 
is even and 
where
. For every
, 
contain exactly two edges of the length 
as 
and 
where
and
By the definition of,
,
and
, we have 
Then the graph 
has an orthogonal 
-labelling with respect to
. By Theorem 3, there exists a CODC of 
by 
with respect to
.
4. Conclusion
In this paper we are concerned with the orthogonal labelling of CODCs of finite regular circulant graphs. We constructed CODCs by certain classes of graphs such as complete bipartite graph, the union of the co-cycles graph with a star, the center vertex of which belongs to the co-cycles graph and graphs that are connected by a one vertex. Finally we introduced general CODCs of the circulant graph.