# Homomorphisms of Cayley graphs and Cycle Double Covers

## TL;DR

This paper investigates a conjecture relating Cayley graph homomorphisms to graph flows, demonstrating partial results and implications for cycle double covers, thus advancing understanding in algebraic graph theory.

## Contribution

It introduces a strengthened conjecture, provides partial proofs for specific cases, and links the conjecture to cycle double cover existence.

## Key findings

- A natural strengthening of the conjecture does not hold universally.
- The original conjecture implies the existence of small oriented cycle double covers.
- Partial results support the conjecture for certain subclasses.

## Abstract

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 an (M, B)-flow has an (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 an oriented cycle double cover with a small number of cycles.