Orthogonal double covers of Cayley graphs

Abstract

Let X and G be graphs, such that G is isomorphic to a subgraph of X . An orthogonal double cover (ODC) of X by G is a collection B = { P ( x ) : x ∈ V ( X ) } of subgraphs of X , all isomorphic with G , such that (i) every edge of X occurs in exactly two members of B and (ii) P ( x ) and P ( y ) share an edge if and only if x and y are adjacent in X . The main question is: given the pair ( X , G ) , is there an ODC of X by G ? An obvious necessary condition is that X is regular. A technique to construct ODCs for Cayley graphs is introduced. It is shown that for all ( X , G ) where X is a 3-regular Cayley graph on an abelian group there is an ODC, a few well known exceptions apart.