%0 Journal Article
%T On Cartesian Products of Orthogonal Double Covers
%A R. El Shanawany
%A M. Higazy
%A A. El Mesady
%J International Journal of Mathematics and Mathematical Sciences
%D 2013
%I Hindawi Publishing Corporation
%R 10.1155/2013/265136
%X 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 ˇ°symmetric startersˇ± (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
%U http://www.hindawi.com/journals/ijmms/2013/265136/