# On Cartesian Products of Orthogonal Double Covers

Authors: R. El Shanawany; M. Higazy; A. El Mesady
Journal: International Journal of Mathematics and Mathematical Sciences
 (2013)

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## Abstract

LetH be a graph on n vertices and 𝒢 a collection of n
subgraphs of H, one for each vertex, where 𝒢 is an orthogonal double
cover (ODC) of H if every edge of H occurs in exactly two members of 𝒢 and any two members share an edge whenever the corresponding
vertices are adjacent in H and share no edges whenever the corresponding
vertices are nonadjacent in H. In this paper, we are concerned with the Cartesian product of symmetric
starter vectors of orthogonal double covers of the complete bipartite graphs
and using this method to construct ODCs by new disjoint unions of complete
bipartite graphs.


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## Body

## 1. Introduction

For the definition of an orthogonal double cover (ODC) of the complete graphKn by a graph G and for a survey on this topic, see [1]. In [2], this concept has been generalized to ODCs of any graph H by a graph G.While in principle any regular graph is worth considering (e.g., the remarkable case of hypercubes has been investigated in [2]), the choice of H=Kn,n is quite natural, and also in view of a technical motivation, ODCs of such graphs are a helpful tool for constructing ODCs of Kn (see [3, page 48]).In this paper, we assumeH=Kn,n, the complete bipartite graph with partition sets of size n each.An ODC ofKn,n is a collection 𝒢={G0,G1,…,Gn-1,F0,F1,…,Fn-1} of 2n subgraphs (called pages) of Kn,n such that(i)
every edge ofKn,n is in exactly one page of {G0,G1,…,Gn-1} and in exactly one page of {F0,F1,…,Fn-1};(ii)
fori,j∈{0,1,2,…,n-1} and i≠j,E(Gi)∩E(Gj)=E(Fi)∩E(Fj)=∅; and |E(Gi)∩E(Fj)|=1 for all i,j∈{0,1,2,…,n-1}.If all the pages are isomorphic to a given graphG, then 𝒢 is said to be an ODC of Kn,n by G.Denote the vertices of the partition sets ofKn,n by {00,10,…,(n-1)0} and {01,11,…,(n-1)1}. The length of an edge x0y1 of Kn,n is defined to be the difference y-x, where x,y∈ℤn={0,1,2,…,n-1}. Note that sums and differences are calculated in ℤn(i.e., sums and differences are calculated modulo n).Throughout the paper we make use of the usual notation:Km,n for the complete bipartite graph with partition sets of sizes m and n, Pn for the path on n vertices, Cn for the cycle on n vertices, Kn for the complete graph on n vertices, K1 for an isolated vertex, G∪H for the disjoint union of  G and H, and mG for m disjoint copies of G.An algebraic construction of ODCs via “symmetric starters” (see Section2) has been exploited to get a complete classification of ODCs of Kn,n by G for n≤9, a few exceptions apart, all graphs G are found this way (see [3, Table 1]). This method has been applied in [3, 4] to detect some infinite classes of graphs G for which there are ODCs of Kn,n by G.In [5], Scapellato et al. studied the ODCs of Cayley graphs and they proved the following. (i) All 3-regular Cayley graphs, except K4, have ODCs by P4. (ii) All 3-regular Cayley graphs on Abelian groups, except K4, have ODCs by P3∪K2. (iii) All 3-regular Cayley graphs on Abelian groups, except K4 and the 3-prism (Cartesian product of C3 and K2), have ODCs by 3K2.Much research on this subject focused on the detection of ODCs with pages isomorphic to a given graphG. For a summary of results on ODCs, see [1, 4]. The other terminologies not defined here can be found in [6].


## 2. Symmetric Starters

All graphs here are finite, simple, and undirected. LetΓ={γ0,…,γn-1} be an (additive) abelian group of order n. The vertices of Kn,n will be labeled by the elements of Γ×ℤ2. Namely, for (v,i)∈Γ×ℤ2 we will write vi for the corresponding vertex and define {wi,uj}∈E(Kn,n)if and only if i≠j, for all w,u∈Γ and i,j∈ℤ2. If there is no chance of confusion, (w,u) will be written instead of {w0,u1} for the edge between the vertices w0,u1.LetG be a spanning subgraph of Kn,n and let a∈Γ. Then the graph G+a with E(G+a)={(u+a,v+a):(u,v)∈E(G)}is called the a-translate of G. The length of an edge e=(u,v)∈E(G)is defined by d(e)=v-u.G is called a half starter with respect to Γ if |E(G)|=n and the lengths of all edges in G are mutually distinct; that is, {d(e):e∈E(G)}=Γ. The following three results were established in [3].Theorem 1.
IfG is a half starter, then the union of all translates of G forms an edge decomposition of Kn,n; that is, ⋃a∈ΓE(G+a)=E(Kn,n).Hereafter, a half starterG will be represented by the vector v(G)=(vγ0,…,vγn-1), where vγi∈Γand (vγi)0is the unique vertex ((vγi,0)∈Γ×{0})that belongs to the unique edge of length γi in G.Two half starter vectorsv(G0)and v(G1)are said to be orthogonal if {vγ(G0)-vγ(G1):γ∈Γ}=Γ.Theorem 2.
If two half starter vectorsv(G0)and v(G1)are orthogonal, then G={Ga,i:(a,i)∈Γ×ℤ2}with Ga,i=Gi+a is an ODC of Kn,n.The subgraphGs of Kn,n with E(Gs)={(u0,v1):(v0,u1)∈E(G)}is called the symmetric graph of G. Note that if G is a half starter, then Gs is also a half starter.A half starterG is called a symmetric starter with respect to Γ if v(G)and v(Gs)are orthogonal.Theorem 3.
Letn be a positive integer and let G be a half starter represented by the vector v(G)=(vγ0,…,vγn-1). Then G is symmetric starter if and only if {vγ-v-γ+γ:γ∈Γ}=Γ.The above results on ODCs of graphs motivated us to consider ODCs ofKmn,mn if we have the ODCs of Kn,n by G and ODCs of Km,m by H where G,H are symmetric starters. In this paper, we have settled the existence problem of ODCs of Kmn,mn by few infinite families of graphs presented in the next section.


## 3. The Main Results

In the following, if there is no danger of ambiguity, if(i,j)∈ℤn×ℤm we can write (i,j) as ij.Theorem 4.
The Cartesian product of any two symmetric starter vectors is a symmetric starter vector with respect to the Cartesian product of the corresponding groups.Proof.
Letv(G)=(v0,v1,…,vn-1)∈ℤnn be a symmetric starter vector of an ODC of Kn,n by G with respect to ℤn, then
(1){vi-v-i+i:i∈ℤn}=ℤn.
Let u(H)=(u0,u1,…,um-1)∈ℤmm  be a symmetric starter vector of an ODC of Km,m by H with respect to ℤm, then
(2){uj-u-j+j:j∈ℤm}=ℤm.
Thenv(G)×u(H)=(v0u0,v0u1,…,viuj,…,vn-1um-1)) where i∈ℤn and j∈ℤm.
From (1) and (2), we conclude
(3){viuj-v-iu-j+ij:ij∈ℤn×ℤm}={(vi-v-i+i)(uj-u-j+j):i∈ℤn,j∈ℤm,ij∈ℤn×ℤm}=ℤn×ℤm.
Thenv(G)×u(H) is a symmetric starter vector of an ODC of Kmn,mn, with respect to ℤn×ℤm, by a new graph G×H which can be described as follows.
SinceE(G)={(vi,vi+i):i∈ℤn} and E(H)={(uj,uj+j):j∈ℤm}, then E(G×H)={(viuj,viuj+ij):ij∈ℤn×ℤm}. It should be noted that G×His not the usual Cartesian product of the graphs G and H that has been studied widely in the literature.All our results based on the following two major points:(1)
the cartesian product construction in Theorem4,(2)
The existence of symmetric starters for a few classes of graphs that can be used as ingredients for cartesian product construction to obtain new symmetric starters. These are as follows.
(1)
K1,n which is a symmetric starter of an ODC of Kn,n whose vector is v(K1,n)=(0,0,0,…,0︷ntimes)∈ℤnn, see Corollary 2.2.7 in [7].(2)
mK2,2 which is a symmetric starter of an ODC of K4m,4m whose vector is v(mK2,2)=(0,1,2,…,2m-1,0,1,2,…,2m-1)∈ℤ4m4m, see [7, Lemma 2.2.13].(3)
K1,2∪K1,2(n-1) which is a symmetric starter of an ODC of  K2n,2n whose vector is v(K1,2∪K1,2(n-1))=(0,n,n,…,n,n︷(n-1)times,0,n,n,…,n,n︷(n-1)times)∈ℤ2n2n,n≥2, and it is easily checked that vi(K1,2∪K1,2(n-1))=v-i(K1,2∪K1,2(n-1)), and hence {vi-v-i+i:i∈ℤ2n}=ℤ2n.(4)
K1,2∪K2,n-1 which is a symmetric starter of an ODC of K2n,2n whose vector is v(K1,2∪K2,n-1)=(0,1,1,1,…,1,1︷(n-1)times,n,n+1,n+1,…,n+1︷(n-1)times)∈ℤ2n2n,n≥2, for this vector, and it is easily checked that
(4)vi-v-i+i={iifi=0,n,orn+iotherwise
and hence {vi-v-i+i:i∈ℤ2n}=ℤ2n.(5)
K2,3 which is a symmetric starter of an ODC of K6,6 whose vector is v(K2,3)=(0,0,3,3,3,0)∈ℤ66, see [4, Theorem 2.2.7].These known symmetric starters will be used as ingredients for the cartesian product construction to obtain new symmetric starters.Theorem 5.
For all positive integersm,n with gcd(m,3)=1, there exists an ODC of K4mn,4mn by mK2,2n∪2m(3n-1)K1.Proof.
Sincev(mK2,2) and v(K1,n) are symmetric starter vectors, then v(mK2,2)×v(K1,n) is a symmetric starter vector with respect to ℤ4m×ℤn (Theorem 4). The resulting symmetric starter graph has the following edges set:
(5)E(mK2,2n)=⋃i=0m-1{(0i,β(2i)),(0i,2β(m+i)),(0(m+i),β(2i)),(0(m+i),2β(m+i):0≤β≤n-1}.Lemma 6.
For any positive integerm with gcd(m,3)=1, there exists an ODC of K16m,16m by mK4,4∪24mK1.Proof.
Sincev(K2,2) and v(mK2,2) are symmetric starter vectors, then v(K2,2)×v(mK2,2) is a symmetric starter vector with respect to ℤ4×ℤ4m (Theorem 4), and the resulting symmetric starter graph has the following edges set:(6)E(mK4,4)=⋃l=0m-1{{{(0l,0l+ij):ij∈{0l,0(l+2m),2l,2(l+2m)}}∪{(0(l+m),0(l+m)+ij):ij∈{0(l+m),0(l+3m),2(l+m),2(l+3m)}}}∪{{(1l,1l+ij):ij∈{1l,1(l+2m),3l,3(l+2m)}}∪{(1(l+m),1(l+m)+ij):ij∈{1(l+m),1(l+3m),3(l+m),3(l+3m)}}}}.Lemma 7.
For any positive integerm with gcd(m,3)=1, there exists an ODC of K32m,32m by 2mK4,4∪48mK1.Proof.
Sincev(2K2,2) and v(mK2,2) are symmetric starter vectors, then v(2K2,2)×v(mK2,2) is a symmetric starter vector with respect to ℤ8×ℤ4m (Theorem 4), and the resulting symmetric starter graph has the following edges set:(7)E(2mK4,4)=⋃l=0m-1{{{(0l,0l+ij):ij∈{0l,0(l+2m),2l,2(l+2m)}}∪{(0(l+m),0(l+m)+ij):ij∈{0(l+m),0(l+3m),2(l+m),2(l+3m)}}}∪{{(2l,2l+ij):ij∈{1l,1(l+2m),7l,7(l+2m)}}∪{(2(l+m),2(l+m)+ij):ij∈{1(l+m),1(l+3m),7(l+m),7(l+3m)}}}∪{{(4l,4l+ij):ij∈{4l,4(l+2m),6l,6(l+2m)}}∪{(4(l+m),4(l+m)+ij):ij∈{4(l+m),4(l+3m),6(l+m),6(l+3m)}}}∪{{(6l,6l+ij):ij∈{3l,3(l+2m),5l,5(l+2m)}}∪{(6(l+m),6(l+m)+ij):ij∈{3(l+m),3(l+3m),5(l+m),5(l+3m)}}}}.The following conjecture generalizes Lemmas6 and 7.Conjecture 8.
For all positive integersm,n with gcd(m,3)=1 and gcd(n,3)=1, there exists an ODC of K16mn,16mn by nmK4,4∪24mnK1.Theorem 9.
For all positive integersm,n≥2, there exists an ODC of K2mn,2mn by K1,2m∪K1,2m(n-1)∪2(mn-1)K1.Proof.
Sincev(K1,m) and v(K1,2∪K1,2(n-1)) are symmetric starter vectors, then v(K1,m)×v(K1,2∪K1,2(n-1)) is a symmetric starter vector with respect to ℤm×ℤ2n (Theorem 4), and the resulting symmetric starter graph has the following edges set:
(8)E(K1,2m∪K1,2m(n-1))=⋃α=0m-1{(00,αβ),(0n,αγ):β∈{0,n},γ∈ℤ2n∖{n}}.Theorem 10.
For all positive integersm,n≥2, there exists an ODC of K4mn,4mn by K1,4∪K1,4(n-1)∪K1,4(m-1)∪K1,4(m-1)(n-1)∪4(mn-1)K1.Proof.
Sincev(K1,2∪K1,2(m-1)) and v(K1,2∪K1,2(n-1)) are symmetric starter vectors, then v(K1,2∪K1,2(m-1))×v(K1,2∪K1,2(n-1)) is a symmetric starter vector with respect to ℤ2m×ℤ2n (Theorem 4), and the resulting symmetric starter graph has the following edges set:
(9)E(K1,4∪K1,4(n-1)∪K1,4(m-1)∪K1,4(m-1)(n-1))=⋃2m-1k=1,k≠m{(00,αβ),(0n,αγ),(m0,kβ),(mn,kγ):α∈{0,m},β∈{0,n},γ∈ℤ2n∖{0,n}}.Theorem 11.
For all positive integersm,n≥2 with gcd(m,3)=1, there exists an ODC of K8mn,8mn by mK2,4∪mK2,4(n-1)∪4m(3n-1)K1.Proof.
Sincev(K1,2∪K1,2(n-1)) and v(mK2,2) are symmetric starter vectors, then v(K1,2∪K1,2(n-1))×v(mK2,2) is a symmetric starter vector with respect to ℤ2n×ℤ4m (Theorem 4), and the resulting symmetric starter graph has the following edges set: (10)E(mK2,4∪mK2,4(n-1))=⋃l=0m-1{{{(0l,0l+ij):ij∈{0l,0(l+2m),2l,2(l+2m)}}∪{(0(l+m),0(l+m)+ij):ij∈{0(l+m),0(l+3m),2(l+m),2(l+3m)}}}∪{{(nl,nl+ij):ij∈{1l,1(l+2m),3l,3(l+2m)}}∪{(n(l+m),n(l+m)+ij):ij∈{1(l+m),1(l+3m),3(l+m),3(l+3m)}}}}.Theorem 12.
For all positive integersm,n≥2, there exists an ODC of K2mn,2mn by K2,m∪K2,m(n-1)∪(3mn-4)K1.Proof.
Sincev(K1,m) and v(K1,2∪K2,n-1) are symmetric starter vectors, then v(K1,m)×v(K1,2∪K2,n-1) is a symmetric starter vector with respect to ℤm×ℤ2n (Theorem 4), and the resulting symmetric starter graph has the following edges set:
(11)E(K2,m∪K2,m(n-1))=⋃α=0m-1{(01,αβ),(00,α0),(0n,α0),(0(n+1),αβ):2≤β≤n}.Lemma 13.
For any positive integerm, there exists an ODC of K6m,6m by K2,3m∪(9m-2)K1.Proof.
Sincev(K1,m) and v(K2,3) are symmetric starter vectors, then v(K1,m)×v(K2,3) is a symmetric starter vector with respect to ℤm×ℤ6 (Theorem 4), and the resulting symmetric starter graph has the following edges set:
(12)E(K2,3m)=⋃α=0m-1{(00,αβ),(03,αβ):β∈{0,1,5}}.


## 4. Conclusion

In conclusion, the known symmetric starters are used as ingredients for the cartesian product construction to obtain new symmetric starters which aremK2,2n∪2m(3n-1)K1,mK4,4∪24mK1,2mK4,4∪48mK1,nmK4,4∪24mnK1,K1,2m∪K1,2m(n-1)∪2(mn-1)K1,K1,4∪K1,4(n-1)∪K1,4(m-1)∪K1,4(m-1)(n-1)∪4(mn-1)K1,mK2,4∪mK2,4(n-1)∪4m(3n-1)K1,K2,m∪K2,m(n-1)∪(3mn-4)K1, and K2,3m∪(9m-2)K1.



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*Source: 265136-2013-04-16.xml*