S^1-flow implies a nowhere-zero 4-flow

Prove that every graph admitting an S^1-flow also admits a nowhere-zero 4-flow.

Background

The paper observes that all currently known classes of graphs admitting S1-flows also admit nowhere-zero 4-flows, which motivates the stated conjecture. The authors note that, using the relationship between S1-flows and nowhere-zero 3-flows, the Four Color Theorem, and Jaeger’s 4-flow theorem, any minimal counterexample would have to be non-planar, non-cubic, and essentially 4-edge-connected, although it could contain many vertices of degree 3.

The conjecture remains unresolved in the paper: the authors explicitly state that they have neither a proof nor a counterexample.

References

We observe that all currently known classes of graphs admitting $S1$-flows also admit nowhere-zero $4$-flows. This motivates the following conjecture.

If a graph $G$ admits an $S1$-flow, then $G$ admits a nowhere-zero $4$-flow.

By Theorem~\ref{thm:3-R3-S1-flow}, the Four Color Theorem, and Jaeger’s $4$-flow theorem, a minimal counterexample, if one exists, must be non-planar, non-cubic, and essentially $4$-edge-connected. It may contain many vertices of degree $3$. At present, we have neither a proof nor a counterexample to this conjecture.

— Reduction Operations and Structural Characterizations of $S^1$-Flows in Graphs  (2608.18725 - Li et al., 19 Aug 2026) in Section 5.2, “Future Directions on Vector Flows and Graph Reductions”