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