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 O is an M(K4)-minor free oriented matroid, then O has a nowhere-zero 3-coflow, i.e., it is 3-colourable in the sense of Hochstättler-Nešetřil. The class of generalized series parallel (GSP) oriented matroids is a class of 3-colourable oriented matroids with no M(K4)-minor. So far, the only technique towards proving that all orientations of a class C of M(K4)-minor free matroids are GSP (and thus 3-colourable), has been to show that every matroid in C has a positive coline. Towards proving Hadwiger's conjecture for the class of gammoids, Goddyn, Hochstättler, and Neudauer conjectured that every gammoid has a positive coline. In this work we disprove this conjecture by showing that there are infinitely many 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.