Homomorphisms of Cayley graphs and cycle double covers (Q5918869)

An \(M\)-flow \(f: E \to M\), where \(M\) is an abelian group is called an \((M, B)\)-flow if \(f(e)\in B\ \forall e\in E\). Here \(B\) is a symmetric subset of \(M\) with non-zero elements. \(f\) is also called a nowhere zero \(M\)-flow. An inspiration for concentrating on such flows is the fact that a planar graph has a proper face \(k\)-coloring if and only if it has a nowhere-zero \(Z_k\)-flow. \textit{M. DeVos} [``A homomorphism problem for flows'', \url{http://www.openproblemgarden.org/op/a_homomorphism_problem_for_flows}] has raised the following conjecture: ``If \(M\), \(M_0\) are abelian groups and \(B\subseteq M\), \(B_0\subseteq M_0\) their symmetric subsets. If there is a graph homomorphism from the Cayley-graph Cay\((M, B)\) into Cayley-graph Cay\((M_0, B_0)\), then every graph with an \((M, B)\)-flow has an \((M_0,B_0)\)-flow''. It is still open. In this paper, the authors develop a new framework under which the equivalence of this conjecture with the conjecture that ``all graphs have homomorphism property'' is proven. Furthermore, they give a new formulation of the homomorphism property called strong homomorphism property and along with a technical definition namely cubification of a digraph \(G\) have established that every non-cubic graph can be reduced to a smaller cubic one. To conclude they show that flows obtained from the homomorphism property or the strong homomorphism property can be transformed into an oriented cycle double cover and deduce that every bridgeless graph has an orientable cycle double cover with at most 6 cycles. The proof technique involved is very new and interesting and opens up new ideas for further exploration.