Let G be a 2-edge-connected graph embedded in the torus. Robertson's conjecture. If the representativity of the embedding is at least 3, then G has a nowhere-zero 4-flow. The paper states that its theorem on cyclically 4-edge-connected toroidal graphs implies this conjecture, so it is solved by the result presented.

Robertson's representativity conjecture for toroidal graphs

The conjecture says that a sufficiently nonseparating embedding of a $2$ -edge-connected toroidal graph forces a nowhere-zero $4$ -flow. The retrieved paper states that its theorem for cyclically $4$ -edge-connected toroidal graphs implies this conjecture.

May 2025 claimed solution

The paper presents the cyclically $4$ -edge-connected toroidal theorem and explicitly says that it implies Robertson’s conjecture. This is a claimed complete resolution, but the scan supplies no independent verification or referee assessment. The same source notes a counterexample in genus $5$ to a broader surface-generalization; that does not refute the toroidal case.

Current status (as of September 2026): Robertson’s toroidal conjecture is claimed solved by the theorem in the 2025 paper, but its correctness and the stated implication remain unverified.

Robertson's representativity conjecture for toroidal graphs

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