IJISET - International Journal of Innovative Science, Engineering & Technology, Vol. 1 Issue 5, July 2014. www.ijiset.com ISSN 2348 – 7968 On Cyclic Orthogonal Double Covers of Circulant Graphs by Certain Graphs R. El-Shanawany and H. Shabana Department of Physics and Engineering Mathematics Faculty of Electronic Engineering - Menoufiya University, Menouf, Egypt. Abstract An orthogonal double cover (ODC) of a graph  is a collection  ={  ∶∈()} of |()| subgraphs (pages) of , such that they cover every edge of H twice and the intersection of any two of them contains exactly one edge. An ODC  of  is cyclic (CODC) if the cyclic group of order |()| is a subgroup of the automorphism group of . In this paper we are concerned with CODC of circulant graphs by a special class of trees and a special class of connected graphs. Keywords Graph decomposition; Cyclic orthogonal double cover; Automorphism group; Orthogonal- labelling. 1. Introduction All graphs we deal with are undirected, finite and simple. Let H be any regular graph, and let  = � 0 ,  1 , ⋯ ,  |()|−1 � be a collection of |()| subgraphs (pages) of . The collection  is an orthogonal double cover (ODC) of  if (i) Every edge of  is contained in exactly two of the pages in , and (ii) For any two distinct pages   and   ∈ , �(  ) ∩ �  �� = 1, if and only if  and  are adjacent in . If all pages are isomorphic to a given graph , then  is an ODC of  by G. According to the obvious properties of ODCs by a graph , the underlying graph  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, 4, 5, 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. 154