Tutte's 3-flow conjecture

Prove that every 4-edge-connected graph admits a nowhere-zero 3-flow.

Background

The paper defines a nowhere-zero 3-flow as an orientation in which the outdegree minus indegree at every vertex is congruent to zero modulo 3. Tutte conjectured that 4-edge-connectivity alone guarantees such an orientation. The paper notes that the conjecture is known for 6-edge-connected graphs, and that proving it for 5-edge-connected graphs would suffice to establish the full conjecture.

References

Famously, Tutte conjectured that planarity is not required in the previous statement:

Every $4$-edge-connected graph admits a nowhere-zero $3$-flow.

— Flow-critical graphs  (2502.01451 - Árnadóttir et al., 3 Feb 2025) in Conjecture (3-flow-conjecture), Section 1, Subsection 1.1

Famously, Tutte conjectured that planarity is not required in the previous statement: Every 4-edge-connected graph admits a nowhere-zero 3-flow.

— Flow-critical graphs  (2502.01451 - Árnadóttir et al., 3 Feb 2025) in Section 1, subsection “Background and context”