A positive coline of a matroid is a coline whose copoint partition has more singular classes than multiple classes. A gammoid is a matroid arising from a directed graph by the gammoid construction. Positive-coline conjecture for gammoids. Every simple gammoid of rank at least two has a positive coline. This conjecture was proposed as a route toward proving the oriented-matroid version of Hadwiger's conjecture, because positive colines imply that all orientations in the relevant minor-closed class are generalized series-parallel. The paper's abstract states that this conjecture is disproved by an infinite class of strict gammoids without positive colines.

Positive-coline conjecture for gammoids

The conjecture, proposed by Goddyn, Hochstättler, and Neudauer, asserts that every simple gammoid of rank at least $2$ has a positive coline. It was proposed in connection with generalized series-parallel orientations and nowhere-zero $3$ -coflows.

Known results

Albrecht and Hochstättler proved the positive-coline claim for rank- $3$ gammoids. A separate result proves that every simple positroid of rank at least $2$ has a weaker “good coline” property.

September 2022 counterexample

On September 14 and 15, 2022, Guzmán-Pro and Hochstättler reported that the conjecture is false: they exhibit an infinite class of strict gammoids without positive colines, via bicircular matroids lacking positive double circuits. The papers also note that all orientations of cobicircular matroids are generalized series-parallel despite this failure. The counterexample claim is unverified in the retrieved sources.

Current status (as of September 2026): The conjecture is claimed false by an infinite counterexample class; no independent verification or contrary result was found.

Positive-coline conjecture for gammoids

Primary source

Progress summary

Solutions 0