Oriented cobicircular matroids are $GSP$

Abstract

Colourings and flows are well-known dual notions in Graph Theory. In turn, the definition of flows in graphs naturally extends to flows in oriented matroids. So, the colour-flow duality gives a generalization of Hadwiger's conjecture about graph colourings, to a conjecture about coflows of oriented matroids. The first non-trivial case of Hadwiger's conjecture for oriented matroids reads as follows. If $\mathcal{O}$ is an $M(K_4)$-minor free oriented matroid, then $\mathcal{O}$ has a now-where $3$-coflow, i.e., it is $3$-colourable in the sense of Hochst\"attler-Ne\v{s}et\v{r}il. The class of generalized series parallel ($GSP$) oriented matroids is a class of $3$-colourable oriented matroids with no $M(K_4)$-minor. So far, the only technique towards proving that all orientations of a class $\mathcal{C}$ of $M(K_4)$-minor free matroids are $GSP$ (and thus $3$-colourable), has been to show that every matroid in $\mathcal{C}$ has a positive coline. Towards proving Hadwiger's conjecture for the class of gammoids, Goddyn, Hochst\"attler, and Neudauer conjectured that every gammoid has a positive coline. In this work we disprove this conjecture by exhibiting an infinite class of strict gammoids that do not have positive colines. We conclude by proposing a simpler technique for showing that certain oriented matroids are $GSP$. In particular, we recover that oriented lattice path matroids are $GSP$, and we show that oriented cobicircular matroids are $GSP$.