Double circuits in bicircular matroids

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.