• Views 656
• Citations 3
• ePub 39
• PDF 387
`International Journal of Mathematics and Mathematical SciencesVolume 2013, Article ID 265136, 4 pageshttp://dx.doi.org/10.1155/2013/265136`
Research Article

On Cartesian Products of Orthogonal Double Covers

Department of Physics and Engineering Mathematics, Faculty of Electronic Engineering, Menoufiya University, Menouf 32952, Egypt

Received 9 December 2012; Revised 11 February 2013; Accepted 18 March 2013

Academic Editor: Ilya M. Spitkovsky

Copyright © 2013 R. El Shanawany et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let be a graph on vertices and a collection of subgraphs of , one for each vertex, where is an orthogonal double cover (ODC) of if every edge of occurs in exactly two members of and any two members share an edge whenever the corresponding vertices are adjacent in and share no edges whenever the corresponding vertices are nonadjacent in . 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.

1. Introduction

For the definition of an orthogonal double cover (ODC) of the complete graph by a graph and for a survey on this topic, see [1]. In [2], this concept has been generalized to ODCs of any graph by a graph .

While in principle any regular graph is worth considering (e.g., the remarkable case of hypercubes has been investigated in [2]), the choice of is quite natural, and also in view of a technical motivation, ODCs of such graphs are a helpful tool for constructing ODCs of (see [3, page 48]).

In this paper, we assume , the complete bipartite graph with partition sets of size each.

An ODC of is a collection of subgraphs (called pages) of such that(i)every edge of is in exactly one page of and in exactly one page of ;(ii)for and ; and for all .

If all the pages are isomorphic to a given graph , then is said to be an ODC of by .

Denote the vertices of the partition sets of by and . The length of an edge of is defined to be the difference , where . Note that sums and differences are calculated in (i.e., sums and differences are calculated modulo ).

Throughout the paper we make use of the usual notation: for the complete bipartite graph with partition sets of sizes and , for the path on vertices, for the cycle on vertices, for the complete graph on vertices, for an isolated vertex, for the disjoint union of and , and for disjoint copies of .

An algebraic construction of ODCs via “symmetric starters” (see Section 2) has been exploited to get a complete classification of ODCs of by for , a few exceptions apart, all graphs are found this way (see [3, Table ]). This method has been applied in [3, 4] to detect some infinite classes of graphs for which there are ODCs of by .

In [5], Scapellato et al. studied the ODCs of Cayley graphs and they proved the following. (i) All -regular Cayley graphs, except , have ODCs by . (ii) All -regular Cayley graphs on Abelian groups, except , have ODCs by . (iii) All -regular Cayley graphs on Abelian groups, except and the -prism (Cartesian product of and ), have ODCs by .

Much research on this subject focused on the detection of ODCs with pages isomorphic to a given graph . 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. Let be an (additive) abelian group of order . The vertices of will be labeled by the elements of . Namely, for we will write for the corresponding vertex and define if and only if , for all and . If there is no chance of confusion, will be written instead of for the edge between the vertices .

Let be a spanning subgraph of and let . Then the graph with is called the a-translate of . The length of an edge is defined by .

is called a half starter with respect to if and the lengths of all edges in are mutually distinct; that is, . The following three results were established in [3].

Theorem 1. If is a half starter, then the union of all translates of forms an edge decomposition of ; that is, .

Hereafter, a half starter will be represented by the vector , where and is the unique vertex that belongs to the unique edge of length in .

Two half starter vectors and are said to be orthogonal if .

Theorem 2. If two half starter vectors and are orthogonal, then with is an ODC of .

The subgraph of with is called the symmetric graph of . Note that if is a half starter, then is also a half starter.

A half starter is called a symmetric starter with respect to if and are orthogonal.

Theorem 3. Let be a positive integer and let be a half starter represented by the vector . Then is symmetric starter if and only if .

The above results on ODCs of graphs motivated us to consider ODCs of if we have the ODCs of by and ODCs of by where are symmetric starters. In this paper, we have settled the existence problem of ODCs of 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 we can write as .

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. Let be a symmetric starter vector of an ODC of by with respect to , then Let be a symmetric starter vector of an ODC of by with respect to , then
Then where and .
From (1) and (2), we conclude
Then is a symmetric starter vector of an ODC of , with respect to , by a new graph which can be described as follows.
Since and , then . It should be noted that is not the usual Cartesian product of the graphs and that has been studied widely in the literature.

All our results based on the following two major points:(1)the cartesian product construction in Theorem 4,(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) which is a symmetric starter of an ODC of whose vector is , see Corollary in [7].(2) which is a symmetric starter of an ODC of whose vector is , see [7, Lemma ].(3) which is a symmetric starter of an ODC of whose vector is , and it is easily checked that , and hence .(4) which is a symmetric starter of an ODC of whose vector is , for this vector, and it is easily checked that and hence .(5) which is a symmetric starter of an ODC of whose vector is , see [4, Theorem ].

These known symmetric starters will be used as ingredients for the cartesian product construction to obtain new symmetric starters.

Theorem 5. For all positive integers with , there exists an ODC of by .

Proof. Since and are symmetric starter vectors, then is a symmetric starter vector with respect to (Theorem 4). The resulting symmetric starter graph has the following edges set:

Lemma 6. For any positive integer with , there exists an ODC of by .

Proof. Since and are symmetric starter vectors, then is a symmetric starter vector with respect to (Theorem 4), and the resulting symmetric starter graph has the following edges set:

Lemma 7. For any positive integer with , there exists an ODC of by .

Proof. Since and are symmetric starter vectors, then is a symmetric starter vector with respect to (Theorem 4), and the resulting symmetric starter graph has the following edges set:

The following conjecture generalizes Lemmas 6 and 7.

Conjecture 8. For all positive integers with and , there exists an ODC of by .

Theorem 9. For all positive integers , there exists an ODC of by .

Proof. Since and are symmetric starter vectors, then is a symmetric starter vector with respect to (Theorem 4), and the resulting symmetric starter graph has the following edges set:

Theorem 10. For all positive integers , there exists an ODC of by .

Proof. Since and are symmetric starter vectors, then is a symmetric starter vector with respect to (Theorem 4), and the resulting symmetric starter graph has the following edges set:

Theorem 11. For all positive integers with , there exists an ODC of by .

Proof. Since and are symmetric starter vectors, then is a symmetric starter vector with respect to (Theorem 4), and the resulting symmetric starter graph has the following edges set:

Theorem 12. For all positive integers , there exists an ODC of by .

Proof. Since and are symmetric starter vectors, then is a symmetric starter vector with respect to (Theorem 4), and the resulting symmetric starter graph has the following edges set:

Lemma 13. For any positive integer , there exists an ODC of by .

Proof. Since and are symmetric starter vectors, then is a symmetric starter vector with respect to (Theorem 4), and the resulting symmetric starter graph has the following edges set:

4. Conclusion

In conclusion, the known symmetric starters are used as ingredients for the cartesian product construction to obtain new symmetric starters which are , and .

References

1. H.-D. O. F. Gronau, M. Grüttmüller, S. Hartmann, U. Leck, and V. Leck, “On orthogonal double covers of graphs,” Designs, Codes and Cryptography, vol. 27, no. 1-2, pp. 49–91, 2002.
2. S. Hartmann and U. Schumacher, “Orthogonal double covers of general graphs,” Discrete Applied Mathematics, vol. 138, no. 1-2, pp. 107–116, 2004.
3. R. El-Shanawany, Hans-D.O. F. Gronau, and M. Grüttmüller, “Orthogonal double covers of ${K}_{n,n}$ by small graphs,” Discrete Applied Mathematics, vol. 138, no. 1-2, pp. 47–63, 2004.
4. M. Higazy, A study of the suborthogonal double covers of complete bipartite graphs [Ph.D. thesis], Menoufiya University, 2009.
5. R. Scapellato, R. El-Shanawany, and M. Higazy, “Orthogonal double covers of Cayley graphs,” Discrete Applied Mathematics, vol. 157, no. 14, pp. 3111–3118, 2009.
6. R. Balakrishnan and K. Ranganathan, A Textbook of Graph Theory, Universitext, chapter 1, Springer, New York, NY, USA, 2nd edition, 2012.
7. R. El-Shanawany, Orthogonal double covers of complete bipartite graphs [Ph.D. thesis], Universität Rostock, 2002.