Homomorphisms of Cayley graphs and Cycle Double Covers

We study the following conjecture of Matt DeVos: If there is a graph homomorphism from Cayley graph Cay(M, B) to another Cayley graph Cay(M′, B′) then every graph with (M, B)-flow has (M′, B′)-flow. This conjecture was originally motivated by the flow-tension duality. We show that a natural strengthening of this conjecture does not hold in all cases but we conjecture that it still holds for an interesting subclass of them and we prove a partial result in this direction. We also show that the original conjecture implies the existence of oriented cycle double cover with a small number of cycles.