Double circuits in bicircular matroids

TL;DR

This paper disproves a conjecture that all orientations of gammoid matroids have positive double circuits, by providing a class of bicircular matroids that lack such circuits, impacting the understanding of matroid colorability.

Contribution

It introduces a large class of bicircular matroids that do not contain positive double circuits, countering previous conjectures about gammoid orientations.

Findings

Bicircular matroids can lack positive double circuits.

Counterexample to the conjecture on gammoid orientations.

Implications for matroid colorability theories.

Abstract

The first non-trivial case of Hadwiger's conjecture for oriented matroids reads as follows. If $\mathcal{O}$ is an $M(K_4)$-free oriented matroid, then $\mathcal{O}$ admits a NZ $3$-coflow, i.e., it is $3$-colourable in the sense of Hochst\"attler-Ne\v{s}et\v{r}il. The class of gammoids is a class of $M(K_4)$-free orientable matroids and it is the minimal minor-closed class that contains all transversal matroids. Towards proving the previous statement for the class of gammoids, Goddyn, Hochst\"attler, and Neudauer conjectured that every gammoid has a positive coline (equivalently, a positive double circuit), which implies that all orientations of gammoids are $3$-colourable. In this brief note we disprove Goddyn, Hochst\"attler, and Neudauers' conjecture by exhibiting a large class of bicircular matroids that do not contain positive double circuits.