On Cartesian Products of Cyclic Orthogonal Double Covers of Circulants

A collection G of isomorphic copies of a given subgraph G of T is said to be orthogonal double cover (ODC) of a graph T by G, if every edge of T belongs to exactly two members of G and any two different elements from G share at most one edge. An ODC G of T is cyclic (CODC) if the cyclic group of order |V(T )| is a subgroup of the automorphism group of G. In this paper, the CODCs of infinite regular circulant graphs by certain infinite graph classes are considered, where the circulant graphs are labelled by the Cartesian product of two abelian groups.


Introduction
A generalization of notion of an orthogonal double cover (ODC) to arbitrary underlying graphs is as follows.Let T be an arbitrary graph with n vertices and let G = {G 0 , G 1 , . . ., G n−1 } be a collection of n spanning subgraphs of T .G is called an orthogonal double cover (ODC) of T if there exists a bijective mapping ϕ : V(T ) → G such that: (1) Every edge of T is contained in exactly two of the graphs G 0 , G 1 , . . ., G n−1 .
(2) For every choice of different vertices a, b of T , Where E(ϕ(a)) and E(ϕ(b)) refer to the edge sets of the graphs ϕ(a) and ϕ(b) respectively, generally E(G) refers to the edge set of the graph G.
Let G be a spanning subgraph of K n,n and let a ∈ Γ.Then the graph G is called a half starter with respect to Γ if |E(G)| = n and the lengths of all edges in G are different, i.e. {d(e) : e ∈ E(G)} = Γ.The following three results were established in (El-Shanawany, 2002).
Theorem 1. (El-Shanawany, 2002) If G is a half starter, then the union of all translates of G forms an edge decomposition of K n,n ,i.e.
Here, the half starter will be represented by the vector: , where v γ i ∈ Γ and (v γ i ) 0 is the unique vertex ((v γ i , 0) ∈ Γ × {0}) that belongs to the unique edge of length γ i .
Theorem 2. (El-Shanawany, 2002) If two half starters v(G 0 ) and v(G 1 ) are orthogonal, then G = {G a,i : (a, i) G is a half starter, then G s is also a half starter.
A half starter G is called a symmetric starter with respect to Γ if v(G) and v(G s ) are orthogonal.
Theorem 3. (El-Shanawany, 2002) Let n be a positive integer and let G be a half starter represented by v  Theorem 6. (El-Shanawany, 2002) Let n be a positive integer.Let G be a symmetric starter of K n,n and let H be the corresponding graph of G. Then H is an orthogonal double cover (ODC) − generating graph with respect to Γ.
Theorem 7. (Gronau et al., 1997).A cyclic orthogonal double covers (CODC) of K n by a graph G exists if and only if there exists an orthogonal labelling of G.

On Cartesian Products of Cyclic Orthogonal Double Covers of Circulants
Hereafter, we will use the operation ⋆ for the usual multiplication and × for cartesian product and if there is no danger of ambiguity, if (i, j) ∈ Z n × Z m we can write (i, j) as i j.The above two Theorems 7, 8 motivated us to the following: Using the fact that there exists a bijective mapping Φ : and hence we consider xy > pq if x > p or if x = p and y > q where xy, pq ∈ Z n 1 × Z n 2 and x ⋆ y, p ⋆ q ∈ Z n 1⋆ n 2 .The circulant graph Circ(n 1 ⋆ n 2 ; X) has a vertex set Z n 1 × Z n 2 , where Z n 1 = {0, 1, . . ., n 1 − 1}, Z n 2 = {0, 1, . . ., n 2 − 1}, and X ⊂ Z n 1 × Z n 2 .Two vertices ab and cd are adjacent if and only if ab − cd = ±(αβ), where αβ ∈ X, and a, c, and α are calculated inside Z n 1 and b, d, and β are calculated inside Z n 2 .For an edge {ab, cd} in Circ(n 1 ⋆ n 2 ; X), the length of {ab, cd} is min{|ab − cd|, n 1 n 2 − |ab − cd|}.Given two edges e 1 = {ab, cd} and e 2 = { f g, uv} of the same length αβ in Circ(n 1 ⋆ n 2 ; X), the rotation-distance r(αβ) between e 1 and e 2 is r(αβ) = min{wz, st : {ab + wz, cd + wz} = e 2 , { f g + st, uv + st} = e 1 }, where addition and difference for a, c, f, and u are calculated inside Z n 1 and for b, d, g, and v are calculated inside Z n 2 .Note that if r(αβ) = αβ, then the edges e 1 and e 2 are adjacent; if r(αβ) αβ, then the edges e 1 and e 2 are nonadjacent.Consider the Cayley graph one of the following cases in the proof of the following theorem is verified.
Theorem 9. A cyclic orthogonal double cover (CODC) of Circ(n 1 ⋆ n 2 ; X) by a graph G exists if and only if there exists an orthogonal X-labelling of G.
Proof.Case 1.Let n 1 be even and n 2 > 1 be odd.
Subcase 1.1.For n 1 > 2, we find that: (a) For every αβ ∈ X 1 : G contains exactly two edges of length αβ, and exactly one edge of length Subcase 1.2.For n 1 = 2, we find that: (a) For every αβ ∈ X 1 : 2 ⌋ }.G contains exactly two edges of length αβ, and exactly one edge of length Case 2. Let n 1 > 1 be odd and n 2 be even.
Subcase 2.1.For n 2 > 2, we find that: (a) For every αβ ∈ X 1 : 1 ≤ α ≤ ⌊ n 1 2 ⌋.G contains exactly two edges of length αβ, and exactly one edge of length Subcase 2.2.For n 2 = 2, we find that: (a) For every αβ ∈ X 1 : G contains exactly two edges of length αβ, and exactly one edge of length Case 3. Let n 1 and n 2 be odd.Subcase 3.1.For n 1 > 3 and n 2 > 3, we find that: (a) For every αβ ∈ X 1 : graph with the following edges set: Theorem 10.For all positive integers m with gcd(m, 3) = 1, there exists a CODC of Circ(16m; X) by H m 1 with respect to Z 4 × Z 4m .

Conclusion
In conclusion, in the future we can get the orthogonal double covers of circulant graphs by new graph classes where the circulant graphs are labelled by the Cartesian product of two abelian groups. Figure1 . A cyclic orthogonal double cover (CODC) of Circ(n; {d 1 , d 2 , . . ., d k }) by a graph G exists if and only if there exists an orthogonal {d 1 , d 2 , . . ., d k }-labelling of G.